Question

Prove the following:

(cos x + cos y)² + (sin x – sin y)² = `4cos^2  ((x + y))/2`

Answer 1

L.H.S. = (cos x + cos y)² + (sin x – sin y)² 

= cos² x + 2 cos x cos y + cos² y + sin²x – 2 sin x sin y + sin² y

= (cos² x + sin² x) + (cos² y + sin² y) + 2(cos x cos y – sin x sin y)

= 1 + 1 + 2 cos (x + y)

= 2 + 2 cos (x + y)

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

= `2 xx 2 cos^2  ((x + y))/2   ...[because 1 + cos theta = 2 cos^2(theta/2)]`

= `4cos^2  ((x + y))/2`

= R.H.S.