Answer
$$\frac{2}{\tan x+\cot x} =\sin 2x$$
Work Step by Step
Since
\begin{align*}
\frac{2}{\tan x+\cot x}&=\frac{2}{\frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}}\\
&=\frac{2}{\frac{\sin^2 x+\cos^2 x}{\cos x\sin x} }\\
&=\frac{2}{\frac{1}{\cos x\sin x} }\\
&=2\sin x\cos x\\
&=\sin 2x
\end{align*}
Then
$$\frac{2}{\tan x+\cot x} =\sin 2x$$
