# Math Help - proving identities problem

1. ## proving identities problem

cos(x+y)cos(x-y) = cos^2x - sin^2y

2. Hello, reino17!

Prove: . $\cos(x+y)\cos(x-y) \;=\; \cos^2\!x - \sin^2\!y$

$\cos(x+y)\cos(x-y) \;=\;(\cos x\cos y - \sin x\sin y)(\cos x\cos y + \sin x\sin y)$

. . . . . . . . . . . . . $=\quad\;\;\cos^2\!x\cos^2\!y \quad - \quad \sin^2\!x\sin^2\!y$

. . . . . . . . . . . . . $=\;\cos^2\!x\overbrace{(1 - \sin^2\!y)} - \overbrace{(1-\cos^2\!x)}\sin^2\!y$

. . . . . . . . . . . . . $=\;\cos^2\!x - \cos^2\!x\sin^2\!y - \sin^2\!y + \cos^2\!x\sin^2\!y$

. . . . . . . . . . . . . $= \qquad \cos^2\!x - \sin^2\!y$

3. this was really helpful. much thanks