# Tan (a/2) equal to ?

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Ujjwal Share
Feb 17, 2018

$\tan \left(\frac{a}{2}\right)$

$= \sin \frac{\frac{a}{2}}{\cos} \left(\frac{a}{2}\right)$ ---(1)

Using the identities

$\sin \left(\frac{a}{2}\right) = \sqrt{\frac{1 - \cos a}{2}}$

And

$\cos \left(\frac{a}{2}\right) = \sqrt{\frac{1 + \cos a}{2}}$

Put these values in (1)

We get , $\tan \left(\frac{a}{2}\right)$

$= \frac{\sqrt{\frac{1 - \cos a}{2}}}{\sqrt{\frac{1 + \cos a}{2}}}$

$= \sqrt{\frac{1 - \cos a}{1 + \cos a}}$

Multiply both numerator and denominator by $1 - \cos a$

$= \sqrt{\frac{1 - \cos a}{1 + \cos a} \times \frac{1 - \cos a}{1 - \cos a}}$

$= \sqrt{{\left(1 - \cos a\right)}^{2} / \left(1 - {\cos}^{2} a\right)}$

We know , $1 - {\cos}^{2} a = {\sin}^{2} a$

= sqrt[(1-cosa)^2 / sin^2a

$= \sqrt{{\left(\frac{1 - \cos a}{\sin} a\right)}^{2}}$

$= \frac{1 - \cos a}{\sin} a$

$= \frac{1}{\sin} a - \cos \frac{a}{\sin} a$

$= \csc a - \cot a$

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