# How do you prove #csc^2xtan^2x-1=tan^2x#?

Prove trig identity

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To prove ( \csc^2(x) \tan^2(x) - 1 = \tan^2(x) ), you can start with the Pythagorean identities for sine and cosine:

[ \sin^2(x) + \cos^2(x) = 1 ]

Then, divide both sides by ( \cos^2(x) ):

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

Using the reciprocal identities for sine and cosine:

[ \csc^2(x) + 1 = \sec^2(x) ]

Rearranging the terms:

[ \csc^2(x) = \sec^2(x) - 1 ]

Now, substitute ( \sec^2(x) = 1 + \tan^2(x) ) (another Pythagorean identity):

[ \csc^2(x) = 1 + \tan^2(x) - 1 ]

[ \csc^2(x) = \tan^2(x) ]

Thus, ( \csc^2(x) \tan^2(x) - 1 = \tan^2(x) ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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