# proving identities problem

#### reino17

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

#### Soroban

MHF Hall of Honor
Hello, reino17!

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

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

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

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

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

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

reino17

#### reino17

this was really helpful. much thanks