Answer
The equation is not an identity.
A counterexample: $\displaystyle \theta=-\frac{\pi}{4}.$
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 $\displaystyle \theta=-\frac{\pi}{4}$, for which
LHS=$\sqrt{2}-(-\sqrt{2})=2\sqrt{2}$
RHS=$\displaystyle \frac{\frac{\sqrt{2}}{2}-(-\frac{\sqrt{2}}{2})}{\frac{\sqrt{2}}{2}(-\frac{\sqrt{2}}{2})}=\frac{\sqrt{2}}{-\frac{1}{2}}=-2\sqrt{2}$
LHS$\neq$RHS, so the equation is not an identity