How do you simplify #(sin^2x-1)/(1+sin^2x)#?

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1. RecognTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}),To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

2. RecognizeTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you canTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

3. Recognize thatTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can useTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

4. Recognize that \To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

5. Recognize that (\To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

6. Recognize that (\sinTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometricTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

7. Recognize that (\sin^To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identitiesTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

8. Recognize that (\sin^2To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities.To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

9. Recognize that (\sin^2xTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approachTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

10. Recognize that (\sin^2x - To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is toTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

11. Recognize that (\sin^2x - 1To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognizeTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

12. Recognize that (\sin^2x - 1)To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize thatTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

13. Recognize that (\sin^2x - 1) canTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that \To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

14. Recognize that (\sin^2x - 1) can beTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sinTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

15. Recognize that (\sin^2x - 1) can be rewrittenTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

16. Recognize that (\sin^2x - 1) can be rewritten as \To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

17. Recognize that (\sin^2x - 1) can be rewritten as (-To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x +To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

18. Recognize that (\sin^2x - 1) can be rewritten as (-\cosTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

19. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cosTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

20. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

21. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x)To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 xTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

22. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) usingTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

23. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the PyTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

24. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the PythagTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). RearTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

25. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity \To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). RearrTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

26. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging thisTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

27. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sinTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identityTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

28. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity,To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

29. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2xTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, weTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

30. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x +To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we getTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

31. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get \To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

32. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cosTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

33. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

34. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2xTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

35. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x =To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x =To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

36. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

37. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1\To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

38. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1). To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

39. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1). 2.To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

40. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).

41. SubstituteTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

42. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).

43. Substitute \To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 xTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

44. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).

45. Substitute (-To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x\To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

46. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).

47. Substitute (-\cosTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). SubTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

48. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).

49. Substitute (-\cos^To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). SubstitTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

50. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).

51. Substitute (-\cos^2To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). SubstitutingTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

52. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).

53. Substitute (-\cos^2xTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting thisTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

54. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).

55. Substitute (-\cos^2x) forTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this intoTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

56. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).

57. Substitute (-\cos^2x) for \To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into theTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

58. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).

59. Substitute (-\cos^2x) for (\To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the givenTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

60. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).

61. Substitute (-\cos^2x) for (\sinTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expressionTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

62. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).

63. Substitute (-\cos^2x) for (\sin^To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

\To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2xTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x -To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\fracTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1)To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sinTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) inTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in theTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numeratorTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 xTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator. To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x -To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator. 3To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator. 3.To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. RewriteTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominatorTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 +To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sinTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 +To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 xTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} =To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2xTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x)To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \fracTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) asTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as \To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sinTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 -To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cosTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2xTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 xTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1)To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x)To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) forTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) -To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easierTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplificationTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification. To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification. 4To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 +To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification. 4.To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. RecognTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 -To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. RecognizeTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize thatTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cosTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that \To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}\To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2xTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

NowTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x)To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now,To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) inTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplifyTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in theTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify theTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numeratorTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numeratorTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator andTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominatorTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sinTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\fracTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x +To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1)To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 -To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) inTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in theTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cosTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominatorTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator canTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancelTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 xTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel eachTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x)To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each otherTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) -To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each other outTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) - To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each other out. To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) - 1To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each other out. 5To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) - 1}}{{To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each other out. 5.To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) - 1}}{{1To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each other out.
5. SimplTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) - 1}}{{1 +To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each other out.
5. SimplifyTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) - 1}}{{1 + (To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each other out.
5. Simplify theTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) - 1}}{{1 + (1To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each other out.
5. Simplify the expressionTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) - 1}}{{1 + (1 -To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each other out.
5. Simplify the expression toTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cosTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each other out.
5. Simplify the expression to \To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each other out.
5. Simplify the expression to (-\To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each other out.
5. Simplify the expression to (-\fracTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 xTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each other out.
5. Simplify the expression to (-\frac{{To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}} =To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each other out.
5. Simplify the expression to (-\frac{{\To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}} = \To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each other out.
5. Simplify the expression to (-\frac{{\cosTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}} = \fracTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each other out.
5. Simplify the expression to (-\frac{{\cos^To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}} = \frac{{1To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each other out.
5. Simplify the expression to (-\frac{{\cos^2xTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}} = \frac{{1 -To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each other out.
5. Simplify the expression to (-\frac{{\cos^2x}}{{To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}} = \frac{{1 - \To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each other out.
5. Simplify the expression to (-\frac{{\cos^2x}}{{\To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}} = \frac{{1 - \cosTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each other out.
5. Simplify the expression to (-\frac{{\cos^2x}}{{\sinTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}} = \frac{{1 - \cos^To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each other out.
5. Simplify the expression to (-\frac{{\cos^2x}}{{\sin^To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}} = \frac{{1 - \cos^2To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each other out.
5. Simplify the expression to (-\frac{{\cos^2x}}{{\sin^2To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}} = \frac{{1 - \cos^2 xTo simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each other out.
5. Simplify the expression to (-\frac{{\cos^2x}}{{\sin^2xTo simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}} = \frac{{1 - \cos^2 x -To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each other out.
5. Simplify the expression to (-\frac{{\cos^2x}}{{\sin^2x +To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}} = \frac{{1 - \cos^2 x - To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each other out.
5. Simplify the expression to (-\frac{{\cos^2x}}{{\sin^2x + To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}} = \frac{{1 - \cos^2 x - 1To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each other out.
5. Simplify the expression to (-\frac{{\cos^2x}}{{\sin^2x + 1To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}} = \frac{{1 - \cos^2 x - 1}}{{To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each other out.
5. Simplify the expression to (-\frac{{\cos^2x}}{{\sin^2x + 1}}To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}} = \frac{{1 - \cos^2 x - 1}}{{1To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each other out.
5. Simplify the expression to (-\frac{{\cos^2x}}{{\sin^2x + 1}}\To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}} = \frac{{1 - \cos^2 x - 1}}{{1 +To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each other out.
5. Simplify the expression to (-\frac{{\cos^2x}}{{\sin^2x + 1}}).To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}} = \frac{{1 - \cos^2 x - 1}}{{1 + To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each other out.
5. Simplify the expression to (-\frac{{\cos^2x}}{{\sin^2x + 1}}).To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}} = \frac{{1 - \cos^2 x - 1}}{{1 + 1To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each other out.
5. Simplify the expression to (-\frac{{\cos^2x}}{{\sin^2x + 1}}).To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}} = \frac{{1 - \cos^2 x - 1}}{{1 + 1 -To simplify (\frac{{\sin^2x - 1}}{{1 + \sin^2x}}):

1. Recognize that (\sin^2x - 1) can be rewritten as (-\cos^2x) using the Pythagorean identity (\sin^2x + \cos^2x = 1).
2. Substitute (-\cos^2x) for (\sin^2x - 1) in the numerator.
3. Rewrite the denominator (1 + \sin^2x) as (\sin^2x + 1) for easier simplification.
4. Recognize that (-\cos^2x) in the numerator and (\sin^2x + 1) in the denominator can cancel each other out.
5. Simplify the expression to (-\frac{{\cos^2x}}{{\sin^2x + 1}}).To simplify (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}), you can use trigonometric identities. One approach is to recognize that (\sin^2 x + \cos^2 x = 1). Rearranging this identity, we get (\sin^2 x = 1 - \cos^2 x). Substituting this into the given expression:

[\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}} = \frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}}]

Now, simplify the numerator and denominator:

[\frac{{(1 - \cos^2 x) - 1}}{{1 + (1 - \cos^2 x)}} = \frac{{1 - \cos^2 x - 1}}{{1 + 1 - \cos^2 x}}] [= \frac{{- \cos^2 x}}{{2 - \cos^2 x}}]

Thus, the simplified form of (\frac{{\sin^2 x - 1}}{{1 + \sin^2 x}}) is (\frac{{- \cos^2 x}}{{2 - \cos^2 x}}).