Answer
See the explanation below.
Work Step by Step
To verify the given identity,
${{\sin }^{2}}\theta \left( 1+{{\cot }^{2}}\theta \right)=1$
Recall Trigonometric Identities,
$\begin{align}
& 1+{{\cot }^{2}}\theta ={{\csc }^{2}}\theta \\
& \csc \theta =\frac{1}{\sin \theta } \\
& \cot \theta =\frac{\cos \theta }{\sin \theta } \\
\end{align}$
Use the above identities and solve the left side of the given expression,
$\begin{align}
& {{\sin }^{2}}\theta \left( 1+{{\cot }^{2}}\theta \right)={{\sin }^{2}}\theta \left( {{\csc }^{2}}\theta \right) \\
& ={{\sin }^{2}}\theta \cdot \frac{1}{{{\sin }^{2}}\theta } \\
& =1
\end{align}$
Therefore,
${{\sin }^{2}}\theta \left( 1+{{\cot }^{2}}\theta \right)=1$
Hence, it is proved that the given identity holds true.
