Question

Establish each identity.

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

Short answer

The right side of the given equation has been simplified to bring the left side.

Step-by-step solution

Step 1. Given Information

The given equation is$\frac{1-2{\cos }^2\theta }{\sin \theta \cos \theta }=\tan \theta -cot\theta$

Step 2. Calculation 

Consider the right side of the given equation and simplify to bring the left side in order to establish the equation.

$$\begin{gathered} \tan \theta -cot\theta =\frac{\sin \theta }{\cos \theta }-\frac{\cos \theta }{\sin \theta } \\ =\frac{{\sin }^2\theta -{\cos }^2\theta }{\sin \theta \cos \theta } \\ =\frac{1-{\cos }^2\theta -{\cos }^2\theta }{\sin \theta \cos \theta } \\ =\frac{1-2{\cos }^2\theta }{\sin \theta \cos \theta } \end{gathered}$$

Hence, Proved