Question

Verifying Trigonometric Identities
1.) cos^2 x-sin^2 x
answer:1-2sin^2 x
2.) cos^2 x-sin^2 x
answer= 2cos^2 x -1
This is confusing. Both problems start the same but equal different things. How would you solve these?

Answer 1

They both start from:
\[\cos^2(x) - \sin^2(x)\] We also know:\[\cos^2(x) + \sin^2(x) = 1\] Now we can either substitute \[1 - \cos^2(x) =\sin^2(x) \]or \[1 - \sin^2(x) = \cos^2(x)\] Depending on which substitution we do, we get either (Substitution 1):\[\cos^2(x) - (1 - \cos^2(x)) = 2\cos^2(x) - 1\]
or (Substitution 2):\[(1 - \sin^2(x)) - \sin^2(x) = 1 - 2\sin^2(x)\]