Half-Angle Tangent Identities

$$\begin{array}{rclrcl} \mr {t\;\isdef \;\tan\left(\frac{\theta}{2}\right)}{\frac{\sin(\theta)}{1+\cos(\theta)}}% {}{\frac{1-\cos(\theta)}{\sin(\theta)}} \mrone {t^2}{=}{\frac{1-\cos(\theta)}{1+\cos(\theta)}}% \mr {\sin(\theta)}{\frac{2t}{1+t^2}}% {\cos(\theta)}{\frac{1-t^2}{1+t^2}}% \mrone {\tan(\theta)}{=}{\frac{2t}{1-t^2}} \end{array}$$