Answer
$\frac{25}{24}$
Work Step by Step
Given expression is-
$\csc(\cos^{-1}\frac{7}{25})$
Assuming $\cos^{-1} \frac{7}{25}$ = $u$ , we get-
$\cos u$ = $\frac{7}{25}$, [$0\leq u\leq \pi$]
From First Pythagorean identity-
$\sin u$ = + $\sqrt {1 - \cos^{2} u}$
= $\sqrt {1 - (\frac{7}{25})^{2}} $
= $\sqrt {1 - \frac{49}{625}} $
= $\sqrt { \frac{625 - 49}{625}} $
= $\sqrt { \frac{576}{625}} $ = $\frac{24}{25}$
i.e. $\sin u$ = $\frac{24}{25}$
Therefore $\csc u$ = $\frac{1}{\sin u}$ = $\frac{1}{\frac{24}{25}}$
i.e. $\csc u$ = $\frac{25}{24}$
i.e. $\csc(\cos^{-1}\frac{7}{25})$ = $\frac{25}{24}$
