Let's solve this problem step by step. We are given that cos x = \(\sqrt{\frac{a}{b}}\) and asked to find the value of cosec x.

Recall that cosec x is the reciprocal of sin x, and that sin x and cos x are related by the Pythagorean identity: sin^2(x) + cos^2(x) = 1. Substituting the given value of cos x into this identity, we get:

sin^2(x) + \(\left(\sqrt{\frac{a}{b}}\right)^2\) = 1

sin^2(x) + \(\frac{a}{b}\) = 1

sin^2(x) = 1 - \(\frac{a}{b}\)

sin^2(x) = \(\frac{b - a}{b}\)

sin(x) = \(\sqrt{\frac{b - a}{b}}\)

Taking the reciprocal of both sides, we get:

cosec(x) = \(\frac{1}{\sqrt{\frac{b - a}{b}}}\)

cosec(x) = \(\sqrt{\frac{b}{b - a}}\)

So, the value of cosec x is **\(\sqrt{\frac{b}{b - a}}\)**.