Answer
10 $|\sec\theta|$
Work Step by Step
Given expression-
$ \sqrt {4x^{2} + 100}$
Substituting $5\tan\theta$ for $x$ as given, the expression becomes-
$ \sqrt {4(5\tan\theta)^{2} + 100}$
= $ \sqrt {4(25\tan^{2}\theta) + 100}$
= $ \sqrt {100\tan^{2}\theta + 100}$
= $ \sqrt {100(\tan^{2}\theta + 1)}$
= $ \sqrt {100 \sec^{2}\theta}$
{ Writing $(\tan^{2}\theta + 1)$ as $ \sec^{2}\theta$ from second Pythagorean identity}
= 10 $|\sec\theta|$
