Answer
(c). 4 $|\sec\theta|$
Work Step by Step
Given expression-
$ \sqrt {x^{2} + 16}$
Also given, $x$ = $4\tan\theta$
Substituting $4\tan\theta$ for $x$, the expression becomes-
$ \sqrt {(4\tan\theta)^{2} + 16}$
= $ \sqrt {16\tan^{2}\theta + 16}$
= $ \sqrt {16(\tan^{2}\theta + 1)}$
= $ \sqrt {16(\sec^{2}\theta)}$
{ Writing $(\tan^{2}\theta + 1)$ as $ \sec^{2}\theta$ from second Pythagorean identity}
= $ \sqrt {16}$.$ \sqrt {\sec^{2}\theta}$
= 4 $|\sec\theta|$
