Chapter 7.4, Problem 1CFU

To determine

To state: the conditions where the three identities of given function is used.

Answer to Problem 1CFU

The identity $\cos 2\theta =1-2{\sin }^2\theta$ is used for writing or solving the equation in terms of sine.

The identity $\cos 2\theta =2{\cos }^2\theta -1$ is used for writing or solving the equation in terms of cosine.

The identity $\cos 2\theta ={\cos }^2\theta -{\sin }^2\theta$ is used when the equation involves eliminating either ${\cos }^2\theta$ or ${\sin }^2\theta$ .

Explanation of Solution

Given information:

  $\cos 2\theta$

Calculation:

The three identities of $\cos 2\theta$ are as follows:

  $\begin{array}{l}\cos 2\theta =1-2{\sin }^2\theta \mathrm{...}\left(1\right)\\ \cos 2\theta =2{\cos }^2\theta -1\mathrm{...}\left(2\right)\\ \cos 2\theta ={\cos }^2\theta -{\sin }^2\theta \mathrm{...}\left(3\right)\end{array}$

The identity (1) is used when the equation needs to be written or solved in terms of sine. The identity (2) is used when the equation needs to be written or solved in terms of cosine. The identity (3) is used when the equation involves eliminating either ${\cos }^2\theta$ or ${\sin }^2\theta$ .