proving identities problem

Soroban

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May 2006
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Lexington, MA (USA)
Hello, reino17!

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

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

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

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

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

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

 
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