$p r o v e t h a t : \frac{s i n 5 x - 2 s i n 3 x + s i n x}{c o s 5 x - c o s x} = t a n x$

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Solution

LHS: $= \frac{s i n 5 x - 2 s i n 3 x + s i n x}{c o s 5 x - c o s x}$

$= \frac{[(s i n 5 x - s i n 3 x) - (s i n 3 x - s i n x)]}{c o s 5 x - c o s x}$

$= \frac{[(2 c o s 4 x s i n x) - (2 c o s 2 x s i n x)]}{- 2 s i n 3 x s i n 2 x}$

$= \frac{2 s i n x (c o s 4 x - c o s 2 x)}{- 2 s i n 3 x .2 s i n x c o s x}$

$= \frac{c o s 4 x - c o s 2 x}{- 2 s i n 3 x c o s x}$

$= \frac{- 2 s i n 3 x s i n x}{- 2 s i n 3 x c o s x}$

$= \frac{s i n x}{c o s x} = t a n x$ = RHS

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