Answer
The equation is not an identity.
A counterexample: $\theta=0.$
Work Step by Step
Graphing utility used: online calculator, desmos.com.
(see below)
The graphs of the two expressions are not equal,
implying that the equation is not an identity.
A counterexample is $\theta=0$, for which
LHS=undefined (cot0 is undefined)
RHS=$\displaystyle \frac{0+1}{1}=1$
LHS$\neq$RHS, so the equation is not an identity