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Basic Trigonometric Identities

Prove the fol...

Question

Prove the following identities
$\frac{s i n 2 α}{1 + c o s 2 α} \cdot \frac{c o s α}{1 + c o s α} = t a n \frac{α}{2}$ .

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Solution

$L H S = \frac{sin 2 α}{1 + cos 2 α} . \frac{cos α}{1 + cos α}$

$= \frac{2 sin α cos α}{2 {cos}^{2} α} . \frac{cos α}{1 + cos α}$

$= \frac{sin α}{1 + cos α}$

$= \frac{2 sin \frac{α}{2} cos \frac{α}{2}}{2 {cos}^{2} \frac{α}{2}}$

$= tan \frac{α}{2} = R H S$

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