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

Question

Verify the following identities. Can you show a procedure please

Answer 1

Proof:

LHS = cos(x +y) cos(x−y)

= 1/2 [cos(x+y+x−y) + cos(x+y-x+y)] (Product-to-Sum Formula)

= 1/2 [cos(2x) + cos(2y)]

= 1/2 [2cos²x − 1 + 1 − 2sin²y] (Double-Angle Formula)

= cos²x − sin²y

= RHS

∴ cos(x +y) cos(x−y) = cos²x − sin²y

Answer 2

cos(x+y)cos(x-y) = (cosxcosy - sinxsiny)(cosxcosy + sinxsiny)

= cos²xcos²y - sin²xsin²y

= cos²x (1 - sin²y) - (1 - cos²x)sin²y

= cos²x - cos²xsin²y - sin²y + cos²xsin²y = cos²x - sin²y