Answer
See the explanation below.
Work Step by Step
$\cot x=\frac{1+\cos 2x}{\sin 2x}$
Consider the left side of the given expression and apply the quotient Identity formula.
$\cot x=\frac{\cos x}{\sin x}$
Multiply and divide by $2\cos x$.
$\begin{align}
& \cot x=\frac{\cos x}{\sin x}\times \frac{2\cos x}{2\cos x} \\
& =\frac{2{{\cos }^{2}}x}{2\sin x\cos x}
\end{align}$
Recall the double angle formula.
$\sin 2x=2\sin x\cos x$
And,
$\begin{align}
& \cos 2x=2{{\cos }^{2}}x-1 \\
& 2{{\cos }^{2}}x=1+\cos 2x \\
\end{align}$
Apply the double angle formula,
$\cot x=\frac{1+\cos 2x}{\sin 2x}$
Hence, it is proved that the given identity $\cot x=\frac{1+\cos 2x}{\sin 2x}$ holds true.
