Double angle identities

An identity means the equation holds true for all values of the unknown(s). An identity is something we prove or use, rather than something we solve.

You previously saw the Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1, and we used it to convert between trig ratios of the same angle.

Double angle identities

sin⁡2θ=2sin⁡θcos⁡θ\sin 2\theta = 2\sin\theta\cos\theta
cos⁡2θ=2cos⁡2θ−1=cos⁡2θ−sin⁡2θ=1−2sin⁡2θ\cos 2\theta = 2\cos^2\theta - 1 = \cos^2\theta - \sin^2\theta = 1-2\sin^2\theta

The three versions of cos⁡2θ\cos 2\theta can be shown be the equivalent using the Pythagorean identity.

The choice of the cos⁡2θ\cos 2\theta depending on whether you want a quadratic in sin⁡θ\sin\theta or cos⁡θ\cos\theta. They are also used to integrate cos⁡2θ\cos^2\theta or sin⁡2θ\sin^2\theta, after some algebra.

Strategies to Proofs

  1. The reading order of your LHS to RHS proof must manipulate only one side of the equation.
  2. The writing order does not have to be the reading order. It is possible to reverse engineer the proof as long as you present it properly.
  3. There are a limited number of identities. Have the formula booklet open to see what might fit.