Chapter 6 - Analytic Trigonometry - Section 6.4 Trigonometric Identities - 6.4 Assess Your Understanding - Page 497: 31

The left hand side is equivalent to $1$ therefore the given equation is an identity. Refer to the solution below.

$\text{ LHS } = \cos^2{\theta}(1+\tan^2{\theta})$ $\text{By Expanding:}$ \begin{align} \text{ LHS } &= \cos^2{\theta}+\cos^2{\theta} \tan^2{\theta} \\[2mm] &= \cos^2{\theta}+\cos^2{\theta} \cdot \dfrac{\sin^2{\theta}}{\cos^2{\theta}} \\[2mm] &= \cos^2{\theta}+\sin^2{\theta} \\[2mm] &= 1 \\[2mm] &= \text{ RHS} \end{align}