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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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