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Question
Prove that: `(cos x + cos y)^2 + (sin x - sin y )^2 = 4 cos^2 (x + y)/2`
Solution
L.H.S. = `(cos x + cos y)^2 + (sin x - sin y )^2 = (2cos (x + y)/2 cos (x - y)/2)^2 + (2 cos (x + y)/2 sin (x - y)/2)^2`
∵ `[ cos A + cos B = 2cos (A + B)/2 cos (A - B)/2, sin A - sin B = 2cos (A + B)/2 sin (A - B)/2]`
= `4 (cos (x + y)/2)^2 [(cos (x - y)/2)^2 + (sin (x + y)/2)^2]`
= `4 (cos (x + y)/2)^2` = R.H.S. [∵ sin2 x + cos2 x = 1]
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