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Question
Prove the following:
cot4x (sin5x + sin3x) = cotx (sin5x − sin3x)
Solution
We have to prove that,
cot4x (sin5x + sin3x) = cot x (sin5x – sin3x)
i.e., to prove that,
`(sin5x + sin3x)/(sin5x - sin3x) = cotx/(cot4x)`
L.H.S. = `(sin5x + sin3x)/(sin5x - sin3x)`
= `(2sin((5x + 3x)/2)*cos((5x - 3x)/2))/(2cos((5x + 3x)/2)*sin((5x - 3x)/2)`
= `(2sin4x*cosx)/(2cos4x*sinx)`
= tan4x · cotx
= `cotx/(cot4x)`
= R.H.S.
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