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Question
Prove the following:
`(sin5x - 2sin3x + sinx)/(cos5x - cosx)` = tanx
Solution
L.H.S. = `(sin5x - 2sin3x + sinx)/(cos5x - cosx)`
= `((sin5x + sinx) - 2sin3x)/(cos5x - cosx)`
= `(2sin((5x + x)/2)*cos((5x - x)/2)-2sin3x)/(-2sin((5x + x)/2)*sin((5x - x)/2)`
= `(2sin3x*cos2x - 2sin3x)/(-2sin3x*sin2x)`
= `(2sin3x(cos2x - 1))/(-2sin3x*sin2x)`
= `(1 - cos2x)/(sin2x)`
= `(2sin^2x)/(2sinx cosx)`
= `sinx/cosx`
= tanx
= R.H.S.
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