Chapter 13.4, Problem 26PPS

To determine

To Prove: $\tan 2\theta =\frac{2}{\cot \theta -\tan \theta }$ is an identity

Answer to Problem 26PPS

  $\tan 2\theta =\frac{2}{\cot \theta -\tan \theta }$ is an identity.

Explanation of Solution

Given information:

We know that, the given expression is $\tan 2\theta =\frac{2}{\cot \theta -\tan \theta }$

Formula used:

  $\tan 2\theta =\frac{2-\tan \theta }{1-{\tan }^2\theta }$

Calculation:

  $\begin{array}{l}\Rightarrow \tan 2\theta =\frac{2}{\cot \theta -\tan \theta }\\ \Rightarrow \tan 2\theta =\frac{2}{\frac{1}{\tan \theta }-\tan \theta }\\ \Rightarrow \tan 2\theta =\frac{2}{\frac{1-{\tan }^2\theta }{\tan \theta }}\\ \Rightarrow \tan 2\theta =\frac{2\tan \theta }{1-{\tan }^2\theta }\\ \Rightarrow \tan 2\theta =\tan 2\theta \\ \therefore \tan 2\theta =\frac{2}{\cot \theta -\tan \theta }isanidentity\end{array}$