Answer
$${\text{We have proved that }}LHS = RHS$$
Work Step by Step
$$\eqalign{
& \tan \theta + \cot \theta = 2\csc 2\theta \cr
& \cr
& LHS = \tan \theta + \cot \theta {\text{ and }}RHS = 2\csc 2\theta \cr
& \cr
& {\text{Use the identity }}\tan \theta = \frac{{\sin \theta }}{{\cos \theta }}{\text{ and cot}}\theta = \frac{{\cos \theta }}{{\sin \theta }} \cr
& LHS = \tan \theta + \cot \theta \cr
& LHS = \frac{{\sin \theta }}{{\cos \theta }} + \frac{{\cos \theta }}{{\sin \theta }} \cr
& LHS = \frac{{{{\sin }^2}\theta + {{\cos }^2}\theta }}{{\cos \theta \sin \theta }} \cr
& LHS = \frac{1}{{\cos \theta \sin \theta }} \cr
& LHS = \frac{2}{{2\cos \theta \sin \theta }} \cr
& LHS = \frac{2}{{\sin 2\theta }} \cr
& LHS = 2\left( {\frac{1}{{\sin 2\theta }}} \right) \cr
& LHS = 2\csc 2\theta \cr
& \cr
& {\text{We have proved that }}LHS = RHS \cr} $$
