1.

Prove the following : (cos x + cos y)2 + (sin x + sin y)2 = 4cos2 (x-y/2)

Answer»

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

= cos2 x + cos2 y + 2cos x.cos y + sin2 x + sin2 y + 2sin x.sin y 

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

= 1 + 1 +2cos(x – y) 

= 2 + 2 cos (x – y) 

= 2[1 + cos(x – y)]

= 2[2cos2 [((\(\frac{x-y}{2}\)))]...[∵ 1 + cos θ = 2 cos2 θ/2]

= 4 cos2 (\(\frac{x-y}{2}\))

=R.H.S



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