How do you solve (cosx+cosy)^2+ (sinx-siny)^2=2+2cos(x+y)?

Answer 1

First use algebra to double distribute

(cosx+cosy)²+(sinx+siny)²

cos²x+2cosxcosy+cos²y+sin²x+2sinxsiny+sin²y

Then the pythagorean identity

cos²x+sin²x+cos²y+sin²y+2cosxcosy+2sinxsiny

1+1+2cosxcosy+2sinxsiny

2+2cosxcosy+2sinxsiny

Last, factor and the angle addition identity for cosine

2+2(cosxcosy+sinxsiny)

2+2cos(x+y) _QED

An important thing to remember with proofs is that you should only work with one side of the equation, you can work with the other side as scrap, but not as part of the actual proof.