Evaluate:

Solution:

(i)

$\cos \left\{\sin ^{-1}\left(-\frac{7}{25}\right)\right\}=\cos \left\{-\sin ^{-1}\left(\frac{7}{25}\right)\right\}$

$=\cos \left\{\sin ^{-1}\left(\frac{7}{25}\right)\right\}$

$=\cos \left\{\cos ^{-1} \sqrt{1-\left(\frac{7}{25}\right)^{2}}\right\}$

$=\cos \left\{\cos ^{-1} \frac{24}{25}\right\}$

$=\frac{24}{25}$

(ii)

$\sec \left\{\cot ^{-1}\left(-\frac{5}{12}\right)\right\}=\sec \left\{\pi-\cot ^{-1}\left(\frac{5}{12}\right)\right\}$

$=-\sec \left\{\cot ^{-1}\left(\frac{5}{12}\right)\right\}$

$=-\sec \left\{\cos ^{-1}\left[\frac{1}{1+\left(\frac{12}{5}\right)^{2}}\right]\right\}$

$=-\sec \left\{\cos ^{-1}\left(\frac{5}{13}\right)\right\}$

$=-\sec \left\{\sec ^{-1}\left(\frac{13}{5}\right)\right\}$

$=-\frac{13}{5}$

(iii)

$\cot \left\{\sec ^{-1}\left(-\frac{13}{5}\right)\right\}=\cot \left\{\sec ^{-1}\left(\pi-\frac{13}{5}\right)\right\}$

$=-\cot \left\{\sec ^{-1}\left(\frac{13}{5}\right)\right\}$

$=-\cot \left\{\tan ^{-1}\left(\frac{\sqrt{1-\left(\frac{5}{13}\right)^{3}}}{\frac{5}{13}}\right)\right\}$

$=-\cot \left\{\tan ^{-1}\left(\frac{12}{5}\right)\right\}$

$=-\cot \left\{\cot ^{-1}\left(\frac{5}{12}\right)\right\}$

$=-\frac{5}{12}$