How do you verify the identity #(sec^2x-1)cos^2x=sin^2x#?

1 Answer
Lil Del
Oct 4, 2016

Simplify the left side of the identity without changing the right side of the identity at all. The left side will simplify to #sin^2x#.

Explanation:

#(sec^2x - 1)cos^2x = sin^2x#
Distribute #cos^2x#:
#sec^2xcos^2x - cos^2x = sin^2x#
Recall that #sec^2x# is defined to be the reciprocal of #cos^2x#, or #1/cos^2x#. Therefore, we now have:
#cos^2x/cos^2x - cos^2x = sin^2x#
This simplifies to:
#1 - cos^2x = sin^2x#
Now recall that #sin^2x + cos^2x = 1#, which can be transformed using the Subtraction Property of Equality to #sin^2x = 1 - cos^2x#. Substituting will then give us:
#sin^2x = sin^2x#
So therefore, the identity has been verified.

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