Prove the following:
$c o t 4 x (s i n 5 x + s i n 3 x) = c o t x (s i n 5 x - s i n 3 x)$

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Solution

$L H S = c o t 4 x (s i n 5 x + s i n 3 x) = \frac{cos 4 x}{sin 4 x} [2 sin (\frac{5 x + 3 x}{2}) cos (\frac{5 x - 3 x}{2})]$
$= 2 cos 4 x cos x = \frac{cos x}{sin x} (2 cos 4 x sin x) = c o t x (s i n 5 x - s i n 3 x)$
Hence proved

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