How do you simplify the expression #cos^2A(sec^2A-1)#?

Answer 1

James Adkins

#cos^2 A (sec^2 A - 1) = sin^2 A#

with exclusion #A != pi/2 + npi# for integer values of #n#.

Take note of this:

#sec A = 1/(cos A)#

#sin^2 A + cos^2 A = 1#

Thus, we discover:

#cos^2 A (sec^2 A - 1) = (cos^2 A)/(cos^2 A) - cos^2 A#

#color(white)(cos^2 A (sec^2 A - 1)) = 1 - cos^2 A#

#color(white)(cos^2 A (sec^2 A - 1)) = sin^2 A#

Note that this identity does not hold for #A = pi/2 + npi#, when #sec A# is undefined, resulting in the left hand side being undefined but the right hand side defined.

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Answer 2

Quinn Anderson

To simplify the expression cos^2(A)(sec^2(A) - 1), first recall the trigonometric identity:

sec^2(A) - 1 = tan^2(A)

Now substitute this identity into the expression:

cos^2(A)(tan^2(A))

Then, recognize the identity:

tan^2(A) = sec^2(A) - 1

Substitute this identity into the expression:

cos^2(A)(sec^2(A) - 1) = cos^2(A)(sec^2(A) - 1) = cos^2(A)tan^2(A)

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Answer from HIX Tutor

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.