Question

Evaluate:
(i) $\cos \left\{{\sin }^{-1}\left(-\frac{7}{25}\right)\right\}$
(ii) $\sec \left\{{\cot }^{-1}\left(-\frac{5}{12}\right)\right\}$
(iii) $\cot \left\{{\sec }^{-1}\left(-\frac{13}{5}\right)\right\}$

Answer 1

(i)
$$\begin{gathered} \cos \left\{{\sin }^{-1}\left(-\frac{7}{25}\right)\right\}=\cos \left\{-{\sin }^{-1}\left(\frac{7}{25}\right)\right\} \\ =\cos \left\{{\sin }^{-1}\left(\frac{7}{25}\right)\right\} \\ =\cos \left\{{\cos }^{-1}\sqrt{1-{\left(\frac{7}{25}\right)}^2}\right\} \\ =\cos \left\{{\cos }^{-1}\frac{24}{25}\right\} \\ =\frac{24}{25} \end{gathered}$$
(ii)
$$\begin{gathered} \sec \left\{{\cot }^{-1}\left(-\frac{5}{12}\right)\right\}=\sec \left\{\mathrm{\pi }-{\cot }^{-1}\left(\frac{5}{12}\right)\right\} \\ =-\sec \left\{{\cot }^{-1}\left(\frac{5}{12}\right)\right\} \\ =-\sec \left\{{\cos }^{-1}\left[\frac{1}{1+{\left({\displaystyle \frac{12}{5}}\right)}^2}\right]\right\} \\ =-\sec \left\{{\cos }^{-1}\left(\frac{5}{13}\right)\right\} \\ =-\sec \left\{{\sec }^{-1}\left(\frac{13}{5}\right)\right\} \\ =-\frac{13}{5} \end{gathered}$$
(iii)
$$\begin{gathered} \cot \left\{{\sec }^{-1}\left(-\frac{13}{5}\right)\right\}=\cot \left\{{\sec }^{-1}\left(\mathrm{\pi }-\frac{13}{5}\right)\right\} \\ =-\cot \left\{{\sec }^{-1}\left(\frac{13}{5}\right)\right\} \\ =-\cot \left\{{\tan }^{-1}\left(\frac{\sqrt{1-{\left({\displaystyle \frac{5}{13}}\right)}^3}}{{\displaystyle \frac{5}{13}}}\right)\right\} \\ =-\cot \left\{{\tan }^{-1}\left(\frac{12}{5}\right)\right\} \\ =-\cot \left\{{\cot }^{-1}\left(\frac{5}{12}\right)\right\} \\ =-\frac{5}{12} \end{gathered}$$