Answer
Counterexample: $\displaystyle \theta=\frac{\pi}{3}$
(sample answer)
Work Step by Step
Use the table of trigonometric functions for special angles (page 128).
For $\displaystyle \theta=\frac{\pi}{3},$
$\tan\theta=\sqrt{3}$
$\displaystyle \cot\theta=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}$
LHS=$\displaystyle \sqrt{3}+\frac{\sqrt{3}}{3}=\frac{3\sqrt{3}+\sqrt{3}}{3}$
$=\displaystyle \frac{4\sqrt{3}}{3}\neq 1$
LHS$\neq$RHS so the identity is not valid.