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Question
Prove that cot 4x (sin 5x + sin 3x) = cot x(sin 5x - sin 3x).
Solution
LHS = cot 4x (sin 5x + sin 3x)
`= cot 4x xx 2 sin ((5x + 3x)/2) cos ((5x - 3x)/2)`
`= 2 (cos 4x)/(sin 4x) xx sin 4x xx cos x` = 2 cos 4x cos x
RHS = cot x (sin 5x - sin 3x)
`= cot x xx 2 sin ((5x - 3x)/2) * cos ((5x + 3x)/2)`
`[sin "A" - sin "B" = 2 sin (("A - B")/2) cos(("A + B")/2)]`
`= (cos x)/cancel(sin x) xx 2 cancel (sin x) cos 4x`
= 2 cos x cos 4x
LHS = RHS.
Hence proved.
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