1.

Prove the following : (cos x – cos y)2 + (sin x – sin y)2 = 4sin2 (\(\frac{x-y}{2}\))

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[2sin2[((\(\frac{x-y}{2}\)))]… [∵ 1 – cos θ = 2 sin2 θ/2]

= 4 sin2(\(\frac{x-y}{2}\))

=R.H.S



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