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Question
Prove that: `((sin 7x + sin 5x) + (sin 9x + sin 3x))/((cos 7x + cos 5x) + (cos 9x + cos 3x)) = tan 6x`
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Solution
L.H.S. = `((sin 7x + sin 5x) + (sin 9x + sin 3x))/((cos 7x + cos 5x) + (cos 9x + cos 3x))`
= `(2sin ((7x + 5x)/2) cos ((7x - 5x)/2) + 2sin ((9x +3x)/2) cos ((9x - 3x)/2))/(2cos ((7x +5x)/2) cos ((7x - 5x)/2) + 2cos ((9x +3x)/2) cos ((9x -3x)/2)`
= `(2[sin6x cosx + sin6x cos3x])/(2[cos6x cos x + cos 6x cos 3x])`
= `(2[cosx + cos3x]sin 6x)/(2[cos x + cos 3x]cos 6x)`
= `(sin 6x)/(cos 6x)`
= tan 6x = R.H.S.
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