Answer
Work on the left side of the identity using $\tan{\theta}=\frac{\sin\theta}{\cos\theta}$ 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:
$\cos\theta(\tan\theta+\cot\theta)=\csc\theta$
Note that
$\tan\theta=\frac{\sin\theta}{\cos\theta}$
$\cot\theta=\frac{\cos\theta}{\sin\theta}$
Work on the left side:
$\cos\theta\left(\dfrac{\sin\theta}{\cos\theta}+\dfrac{\cos\theta}{\sin\theta}\right)\\
=\sin\theta+\dfrac{\cos^2\theta}{\sin\theta}\\
=\dfrac{\sin^2\theta}{\sin\theta}+\dfrac{\cos^2\theta}{\sin\theta}\\
=\dfrac{\sin^2\theta+\cos^2\theta}{\sin\theta}\\
=\dfrac{1}{\sin\theta}\\
=\csc\theta$
