Question

Establish each identity.

$cot2\theta =\frac{co{t}^2\theta -1}{2cot\theta }$

Short answer

Thus the given identity exists.

Step-by-step solution

Step 1. Given

given expression

$cot2\theta =\frac{co{t}^2\theta -1}{2cot\theta }$

Step 2. Concept

we will use standard trigonometric formula's for solving the given expression.

Step 3. Calculation

Consider

$$\begin{gathered} cot2\theta =\frac{1}{\tan 2\theta } \\ U\sin gdouble-angleformula \\ \tan 2\theta =\frac{2\tan \theta }{1-{\tan }^2\theta } \\ Now \\ \frac{1}{\tan 2\theta }=\frac{1-{\tan }^2\theta }{2\tan \theta } \\ \frac{1-{\displaystyle \frac{1}{co{t}^2\theta }}}{{\displaystyle \frac{2}{cot\theta }}}=\frac{co{t}^2\theta -1}{2cot\theta } \end{gathered}$$