Question

Establish the identity

$\tan \theta +cot\theta =2csc\left(2\theta \right)$

Short answer

By converting tan$\theta$ and cot$\theta$ into sin$\theta$ and cos$\theta$

we get the tan$\theta$ + cot$\theta$ =2csc(2$\theta$)

Step-by-step solution

Step 1. Given information  

$\tan \theta +cot\theta =2csc\left(2\theta \right)$

Step 2. Calculations 

$$\begin{gathered} \tan \theta +cot\theta =2csc\left(2\theta \right) \\ TakeLHS,\tan \theta +cot\theta \\ \frac{\sin \theta }{\cos \theta }+\frac{\cos \theta }{\sin \theta }=\frac{{\sin }^2\theta +{\cos }^2\theta }{\cos \theta \sin \theta }\tan \theta =\frac{\sin \theta }{\cos \theta } \\ cot\theta =\frac{\cos \theta }{\sin \theta } \\ =\frac{1}{\cos \theta \sin \theta }=\frac{2}{2\cos \theta \sin \theta }=\frac{2}{\sin 2\theta }2\cos \theta \sin \theta =\sin 2\theta \\ =2csc\left(2\theta \right) \end{gathered}$$

Hence LHS=RHS