Question

Prove the following:

cos(x + y).cos(x − y) = cos²y − sin²x

Answer 1

L.H.S. = cos(x + y) · cos (x – y)

= (cosx cosy – sinx siny) · (cosx cosy + sinx siny)

= cos²x cos²y – sin²x sin²y .....[∵ (a − b) (a + b) = a² − b²]

= (1 – sin²x) cos²y – sin²x (1 – cos²y) ......[∵ sin²θ + cos²θ = 1]

= cos²y – sin²x cos²y – sin²x + sin²x cos²y

= cos²y – sin²x

= R.H.S.