Question

Prove the identity (cosx+cosy)^2+(sinx-siny)^2= 2+2cos(x+y)

If at all possible, please provide a two-columned answer, explaining each step. Thank you :D

Answer 1

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Answer 2

(cos(x) + cos(y))^2 + (sin(x) - sin(y))^2 Remove the brackets

cos^2(x) + cos^2(y) + 2cos(x)*cos(y) + sin^2(x) - 2(sin(x)*sin(y) + sin^2(y) Combine these two in bold to make 1 because sin^2(x) + cos^2(x) = 1

1 + cos^2(y) + 2cos(x)*cos(y) - 2*sin(x)*cos(y) + sin^2(y)
These two in bold also make 1

2 + 2cos(x)*cos(y) - 2*sin(X)*sin(y) Bring out a common factor of 2
2 +2(cos(x)*cos(y) - sin(x)*sin(y) )

but cos(x+y ) = cos(x)*cos(y) - sin(x)*sin(y)

2 + 2* cos(x + y) is your final answer.