Answer
See the explanation below.
Work Step by Step
To verify the given identity,
$\frac{\tan \theta \cot \theta }{\csc \theta }=\sin \theta $
Recall Trigonometric Identities,
$\begin{align}
& \csc \theta =\frac{1}{\sin \theta } \\
& \sec \theta =\frac{1}{\cos \theta } \\
& \tan \theta =\frac{\sin \theta }{\cos \theta } \\
& \cot \theta =\frac{\cos \theta }{\sin \theta } \\
\end{align}$
Use the above identities and solve the left side of the given expression,
$\begin{align}
& \frac{\tan \theta \cot \theta }{\csc \theta }=\frac{\frac{\sin \theta }{\cos \theta }\cdot \frac{\cos \theta }{\sin \theta }}{\frac{1}{\sin \theta }} \\
& =\frac{1}{\frac{1}{\sin \theta }} \\
& =1\cdot \frac{\sin \theta }{1} \\
& =\sin \theta
\end{align}$
Therefore,
$\frac{\tan \theta \cot \theta }{\csc \theta }=\sin \theta $
Hence, it is proved that the given identity holds true.
