# Half Angle Formulas/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.

## Proof

 $\displaystyle \tan \frac \theta 2$ $=$ $\displaystyle \frac {1 - \cos \theta} {\sin \theta}$ Half Angle Formula for Tangent: Corollary 2 $\displaystyle$ $=$ $\displaystyle \frac 1 {\sin \theta} - \frac {\cos \theta} {\sin \theta}$ $\displaystyle$ $=$ $\displaystyle \csc \theta - \cot \theta$ Cosecant is Reciprocal of Sine and Cotangent is Cosine divided by Sine

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

$\blacksquare$