Answer
See the explanation below.
Work Step by Step
$\sin 2t-\cot t=-\cot t\cos 2t$
Recall the Trigonometric Identities and apply below,
$\begin{align}
& \cot t=\frac{\cos t}{\sin t} \\
& \cos 2t=1-2{{\sin }^{2}}t \\
& \sin 2t=2\sin t\cos t \\
\end{align}$
Consider the right side of the given expression and apply the above formula,
$\begin{align}
& -\cot t\cos 2t=-\frac{\cos t}{\sin t}\left( 1-2{{\sin }^{2}}t \right) \\
& =-\frac{\cos t}{\sin t}+\frac{\cos t2{{\sin }^{2}}t}{\sin t} \\
& =-\frac{\sin t}{\cos t}+2\sin t\cos t \\
& =-\cot t+\sin 2t
\end{align}$
Hence, it is proved that the given identity $\sin 2t-\cot t=-\cot t\cos 2t$ holds true.
