Answer
Work on the left side of the identity using $\tan{\theta}=\frac{\sin\theta}{\cos\theta}$, $\sec\theta=\frac{1}{\cos\theta}$, and $\cot\theta=\frac{\cos\theta}{\sin\theta}$.
Refer to the step-by-step part below for the complete proof.
Work Step by Step
We have to show that: $\sin\theta(\cot\theta+\tan\theta)=\sec\theta$
We know that
$\tan\theta=\dfrac{\sin\theta}{\cos\theta}$
$\cot\theta=\dfrac{\cos\theta}{\sin\theta}$
$\sec\theta=\dfrac{1}{\cos\theta}$
Work on the left side of the identity.
Distribute $\sin\theta$ then cancel common factors:
$\sin\theta\left(\dfrac{\cos\theta}{\sin\theta}+\dfrac{\sin\theta}{\cos\theta}\right)$
$=\cos\theta+\dfrac{\sin^2\theta}{\cos\theta}$
Make the expressions similar by using their LCD which is $\cos\theta$, then simplify:
$=\dfrac{\cos^2\theta}{\cos\theta}+\dfrac{\sin^2\theta}{\cos\theta}$
$=\dfrac{\cos^2\theta+\sin^2\theta}{\cos\theta}$
Since $\cos^2\theta+\sin^2\theta=1$, then the expression above simplifies to:
$=\dfrac{1}{\cos\theta}$
$=\sec\theta$
The proof is complete.
