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

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Answer 1 · 2017-01-19T19:34:14

Let's do a little bit of factoring.

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

Use the identity #sin^2theta + cos^2theta = 1#:

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

Secant and cosine are inverses; their product is #1#.

#2 - sin^2x - cos^2x = 1#

You will want to convert all to sine or all to cosine, using the pythagorean identity given above.

#2 - (1 - cos^2x) - cos^2x = 1#

#2 - 1 + cos^2x - cos^2x = 1#

#1 + cos^2x - cos^2x = 1#

#1 = 1#

#LHS = RHS#

The identity is proved.

Hopefully this helps!