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Question
Prove that: sin x + sin 3x + sin 5x + sin 7x = 4 cos x cos 2x sin 4x
Solution
L.H.S. = sin x + sin 3x + sin 5x + sin 7x = (sinx + sin7x) + (sin3x + sin5x)
= `2sin (7x + x)/2 cos (7x - x)/2 + 2sin (5x +3x)/2 cos (5x - 3x)/2` ∵ `[sinx + sin y = 2sin (x + y)/2 cos (x - y)/2]`
= 2sin 4x cos 3x + 2sin 4x cos x [∵ cos (-θ) = cos θ]
= 2sin 4x (cos 3x + cosx) ∵ `[cos x + cos y = 2cos (x + y)/2 cos (x - y)/2]`
= `2sin 4x (2cos (3x +x)/2 cos (3x - x)/2)`
= 2 sin 4x (2cos 2x cos x) = 4 cos x cos 2x sin 4x
= R.H.S.
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