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Question
Prove that `(sin x + sin 3x + sin 5x + sin 7x)/(cos x + cos x + cos 5x cos 7x)` = tan 4x
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Solution
Nr: (sin x + sin 7x) + (sin 3x + sin 5x)
= `[2sin (7x + x)/2 cos (7x - x)/2] + [2sin (5x + 3x)/2 cos (5x - 3x)/2]`
= 2 sin 4x cos 3x + 2 sin 4x cos x
= 2 sin 4x (cos 3x + cos x) .....(1)
Dr. (cos x + cos 7x) + (cos 3x + cos 5x)
= `[2cos (7x + x)/2 cos (7x - x)/2] + [2cos (5x + 3x)/2 cos (5x - 3x)/2]`
= 2 cos 4x cos 3x + 2 cos 4x cosx
= 2 cos 4x (cos 3x + cos x) .....(2)
L.H.S = `((1))/((2))`
= `(2sin 4x(cos 3x + cos x))/(2cos 4x(cos 3x + cos x))`
= tan 4x
= R.H.S
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