Answer
See the explanation below.
Work Step by Step
${{\cos }^{2}}\theta \left( 1+{{\tan }^{2}}\theta \right)=1$
Recall Trigonometric Identities,
$\begin{align}
& 1+{{\tan }^{2}}\theta ={{\sec }^{2}}\theta \\
& \sec \theta =\frac{1}{\cos \theta } \\
\end{align}$
Use the above identities and solve the left side of the given expression,
$\begin{align}
& {{\cos }^{2}}\theta \left( 1+{{\tan }^{2}}\theta \right)={{\cos }^{2}}\theta \left( {{\sec }^{2}}\theta \right) \\
& ={{\cos }^{2}}\theta \cdot \frac{1}{{{\cos }^{2}}\theta } \\
& =1
\end{align}$
Therefore,
${{\cos }^{2}}\theta \left( 1+{{\tan }^{2}}\theta \right)=1$
