Answer
The left side of the equation is equivalent to $\sec^2{x}\tan{x}$ therefore the given equation is an identity.
Refer to the solution below.
Work Step by Step
\begin{align}
\text{LHS } &= \tan^3{x}+\tan{x} \\[2mm]
&= \tan{x}(\tan^2{x}+1) \\[2mm]
&= \tan{x} \sec^2{x} \\[2mm]
&= \text{ RHS}
\end{align}
