Answer
See the explanation below.
Work Step by Step
The expression on the left side $\cos x\left( \tan x+\cot x \right)$ can be written in the form of cos and sin by using the quotient identity.
We know the quotient identity of trigonometry $\cot x=\frac{\cos x}{\sin x}$ and $\tan x=\frac{\sin x}{\cos x}$.
Therefore, the expression on the left side can be written and simplified as:
$\begin{align}
& \text{cos }x\left( \tan x+\cot x \right)=\cos x\left( \frac{\sin x}{\cos x}+\frac{\cos x}{\sin x} \right) \\
& =\cos x\left( \frac{\sin x}{\cos x}.\frac{\sin x}{\sin x}+\frac{\cos x}{\sin x}.\frac{\cos x}{\cos x} \right) \\
& =\cos x\left( \frac{{{\sin }^{2}}x}{\sin x\cos x}+\frac{{{\cos }^{2}}x}{\sin x\cos x} \right) \\
& =\cos x\left( \frac{{{\sin }^{2}}x+{{\cos }^{2}}x}{\sin x\cos x} \right)
\end{align}$
And the expression can be further simplified by applying the Pythagorean identity ${{\sin }^{2}}x+{{\cos }^{2}}x=1$
$\begin{align}
& \cos x\left( \frac{{{\sin }^{2}}x+{{\cos }^{2}}x}{\sin x\cos x} \right)=\cos x.\frac{1}{\sin x\cos x} \\
& =\frac{1}{\sin x} \\
& =\csc x
\end{align}$
And the expression $\frac{1}{\sin x}$ can be further simplified by applying the reciprocal identity $\frac{1}{\sin x}=\csc x$.
