Identity transformations of Trigonometric Expressions.
prove the following identities.
$\frac{s i n α + s i n 5 α}{c o s α + c o s 5 α} = t a n 3 α .$

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Solution

$\frac{s i n α + s i n 5 α}{c o s α + c o s 5 α} = t a n 3 α$
$= \frac{2 s i n \frac{2 + 5 α}{2} . c o s \frac{2 - 5 α}{2}}{2 c o s \frac{2 + 5 α}{2} . c o s \frac{2 - 5 α}{2}}$
$= \frac{2 s i n 3 α c o s 2 α}{2 c o s 3 α . c o s 2 α}$
$= t a n 3 α$

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