Proving Identities – Half angles based on the Double Angle formulae

Some identities work with half angles which are based on the double angle identities.

Double angle identities

  •     \begin{align*} \sin{2A} \equiv 2 \sin A \cos A \end{align} \\

        \begin{align*} \cos {2A} &\equiv \cos {^2{A}} - \sin {^2{A}} \\  &\equiv 2 \cos {^2{A}} - 1  \\  &\equiv 1 - 2 \sin {^2{A}} \end{align*} \\

        \begin{align*} \tan {2A} \equiv \dfrac{2 \tan A}{1 - \tan {^2{A}}} \end{align}

The following examples work with half angles:

Prove:

  1. \dfrac{{\sin \theta }}{{1 + \cos \theta }} \equiv \tan \dfrac{\theta }{2}

Prove:

  1. \mathrm{cosec} \theta  + \cot \theta  \equiv \cot \dfrac{\theta }{2}