cos(x + y) cos(x − y) = cos2 x − sin2 y

Question

I have to show how the problem equals each other by proving the identity.

Answer 1

Assuming that the equation is cos(x + y) cos(x - y) = cos² x - sin² y.

LHS part of the equation is cos(x + y) cos(x - y )

= [cos(x)cos(y) - sin(x)sin(y)] [cos(x)cos(y) + sin(x)sin(y)]

= [cos(x)cos(y) - sin(x)sin(y)] [cos(x)cos(y) + sin(x)sin(y)]

= cos²(x)cos²(y) +cos(x)cos(y)sin(x)sin(y) - sin(x)sin(y)cos(x)cos(y) - sin²(x)sin²(y)

= cos²(x)cos²(y) - sin²(x)sin²(y)

= cos²(x) [1 - sin²(y)] - [1 - cos²(x)] sin²(y)

= cos²(x) - cos²(x) sin²(y) - sin²(y) + sin²(y) cos²(x)

= cos²(x) - sin²(y).

= RHS part of the equation.