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

  • sin 2A ≡ 2sin A cos A
    cos 2A ≡ cos2A – sin2A
    or
    cos 2A ≡ 2cos2A – 1
    or
    cos 2A ≡ 1 – 2sin2A
    tan 2A ≡ (2 tan A) / (1 – tan2A)

Example 1

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}