Half Angle Formula for Tangent/Corollary 3

Theorem

$\tan \dfrac \theta 2 = \csc \theta - \cot \theta$

where $\tan$ denotes tangent, $\csc$ denotes cosecant and $\cot$ denotes cotangent.

When $\theta = k \pi$, the right hand side of this formula is undefined.

$\ds \tan \frac \theta 2$

$=$

$\ds \frac {1 - \cos \theta} {\sin \theta}$

$\ds $

$=$

$\ds \frac 1 {\sin \theta} - \frac {\cos \theta} {\sin \theta}$

$\ds $

$=$

$\ds \csc \theta - \cot \theta$

When $\theta = k \pi$, both $\cot \theta$ and $\csc \theta$ are undefined.

$\blacksquare$