Chapter 5 - Section 5.1 - Verifying Trigonometric Identities - Exercise Set - Page 660: 96

The original identity is $1+{{\tan }^{2}}x=\frac{1}{{{\cos }^{2}}x}$.

Let us consider one of the Pythagorean identities: $1+{{\tan }^{2}}x={{\sec }^{2}}x$ Now, using one of the reciprocal identities, which is $\sec x=\frac{1}{\cos x}$, we get: $\begin{align} & 1+{{\tan }^{2}}x={{\sec }^{2}}x \\ & 1+{{\tan }^{2}}x=\frac{1}{{{\cos }^{2}}x} \\ \end{align}$