Answer
See the explanation below.
Work Step by Step
We use the trigonometric identities, $\tan x=\frac{\sin x}{\cos x}$ and $\cot x=\frac{\cos x}{\sin x}$.
$\tan x+\cot x=\frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}$
Now, multiplying by $\frac{\sin x}{\sin x}$ and $\frac{\cos x}{\cos x}$.
$\begin{align}
& \frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}=\frac{\sin x}{\cos x}\times \frac{\sin x}{\sin x}+\frac{\cos x}{\sin x}\times \frac{\cos x}{\cos x} \\
& =\frac{{{\sin }^{2}}x}{\sin x\cos x}+\frac{{{\cos }^{2}}x}{\sin x\cos x} \\
& =\frac{{{\sin }^{2}}x+{{\cos }^{2}}x}{\sin x\cos x}
\end{align}$
We use the trigonometric identity ${{\sin }^{2}}x+{{\cos }^{2}}x=1$ and multiply the numerator and denominator by 2.
$\begin{align}
& \frac{{{\sin }^{2}}x+{{\cos }^{2}}x}{\sin x\cos x}=\frac{1}{\sin x\cos x} \\
& =\frac{2}{2\sin x\cos x}
\end{align}$
After that, use identities: $2\sin x\cos x=\sin 2x$ and $\frac{1}{\sin x}=\csc x$.
Simplified further,
$\begin{align}
& \frac{2}{2\sin x\cos x}=\frac{2}{\sin 2x} \\
& =2\csc 2x
\end{align}$
Thus, the left side of the equation is equal to $2\csc 2x$.
Hence, the left side is equal to $2\csc 2x$.
