Evaluate each of the following:
(i) $\sin \left(\sin ^{-1} \frac{7}{25}\right)$
(ii) $\sin \left(\cos ^{-1} \frac{5}{13}\right)$
(iii) $\sin \left(\tan ^{-1} \frac{24}{7}\right)$
(iv) $\sin \left(\sec ^{-1} \frac{17}{8}\right)$
(v) $\operatorname{cosec}\left(\cos ^{-1} \frac{3}{5}\right)$
(vi) $\sec \left(\sin ^{-1} \frac{12}{13}\right)$
(vii) $\tan \left(\cos ^{-1} \frac{8}{17}\right)$
(viii) $\cot \left(\cos ^{-1} \frac{3}{5}\right)$
(ix) $\cos \left(\tan ^{-1} \frac{24}{7}\right)$
(i) $\sin \left(\sin ^{-1} \frac{7}{25}\right)=\frac{7}{25}$
(ii)
$\sin \left(\cos ^{-1} \frac{5}{13}\right)=\sin \left[\sin ^{-1} \sqrt{1-\left(\frac{5}{13}\right)^{2}}\right] \quad\left[\because \cos ^{-1} x=\sin ^{-1} \sqrt{1-x^{2}}\right]$
$=\sin \left[\sin ^{-1}\left(\sqrt{1-\frac{25}{169}}\right)\right]$
$=\sin \left[\sin ^{-1}\left(\sqrt{\frac{144}{169}}\right)\right]$
$=\sin \left[\sin ^{-1} \frac{12}{13}\right]$
$=\frac{12}{13}$
$\sin \left(\tan ^{-1} \frac{24}{7}\right)=\sin \left(\sin ^{-1} \frac{\frac{24}{7}}{\sqrt{1+\left(\frac{24}{7}\right)^{2}}}\right) \quad\left[\because \tan ^{-1} x=\frac{x}{\sqrt{1+x^{2}}}\right]$
$=\sin \left(\sin ^{-1} \frac{\frac{24}{7}}{\sqrt{1+\frac{576}{49}}}\right)$
$=\sin \left(\sin ^{-1} \frac{\frac{24}{7}}{\sqrt{\frac{625}{49}}}\right)$
$=\sin \left(\sin ^{-1} \frac{\frac{24}{7}}{\frac{24}{7}}\right)$
$=\frac{24}{25}$
(iv)
$\sin \left(\sec ^{-1} \frac{17}{8}\right)=\sin \left(\cos ^{-1} \frac{8}{17}\right)$
$=\sin \left[\sin ^{-1} \sqrt{1-\left(\frac{8}{17}\right)^{2}}\right] \quad\left[\because \cos ^{-1} x=\sin ^{-1} \sqrt{1-x^{2}}\right]$
$=\sin \left[\sin ^{-1}\left(\sqrt{1-\frac{64}{289}}\right)\right]$
$=\sin \left[\sin ^{-1}\left(\sqrt{\frac{225}{289}}\right)\right]$
$=\sin \left[\sin ^{-1} \frac{15}{17}\right]$
$=\frac{15}{17}$
(v)
$\operatorname{cosec}\left(\cos ^{-1} \frac{3}{5}\right)=\operatorname{cosec}\left[\sin ^{-1} \sqrt{1-\left(\frac{3}{5}\right)^{2}}\right] \quad\left[\because \cos ^{-1} x=\sin ^{-1} \sqrt{1-x^{2}}\right]$
$=\operatorname{cosec}\left[\sin ^{-1}\left(\sqrt{1-\frac{9}{25}}\right)\right]$
$=\operatorname{cosec}\left[\sin ^{-1}\left(\sqrt{\frac{16}{25}}\right)\right]$
$=\operatorname{cosec}\left[\sin ^{-1} \frac{4}{5}\right]$
$=\operatorname{cosec}\left[\operatorname{cosec}^{-1} \frac{5}{4}\right]$
$=\frac{5}{4}$
(vi)
$\sec \left(\sin ^{-1} \frac{12}{13}\right)=\sec \left[\cos ^{-1} \sqrt{1-\left(\frac{12}{13}\right)^{2}}\right] \quad\left[\because \sin ^{-1} x=\cos ^{-1} \sqrt{1-x^{2}}\right]$
$=\sec \left[\cos ^{-1}\left(\sqrt{1-\frac{144}{169}}\right)\right]$
$=\sec \left[\cos ^{-1}\left(\sqrt{\frac{25}{169}}\right)\right]$
$=\sec \left[\cos ^{-1} \frac{5}{13}\right]$
$=\sec \left[\sec ^{-1} \frac{13}{5}\right]$
$=\frac{13}{5}$
(vii)
$\tan \left(\cos ^{-1} \frac{8}{17}\right)=\tan \left\{\tan ^{-1} \frac{\sqrt{1-\left(\frac{8}{17}\right)^{2}}}{\frac{8}{17}}\right\} \quad\left[\because \cos ^{-1} x=\tan ^{-1}\left(\frac{\sqrt{1-x^{2}}}{x}\right)\right]$
$=\tan \left(\tan ^{-1} \frac{\frac{15}{17}}{\frac{8}{17}}\right)$
$=\frac{15}{8}$
(viii)
$\cot \left(\cos ^{-1} \frac{3}{5}\right)=\cot \left\{\tan ^{-1} \frac{\sqrt{1-\left(\frac{3}{5}\right)^{2}}}{\frac{3}{5}}\right\} \quad\left[\because \cos ^{-1} x=\tan ^{-1}\left(\frac{\sqrt{1-x^{2}}}{x}\right)\right]$
$=\cot \left(\tan ^{-1} \frac{\frac{4}{5}}{\frac{3}{5}}\right)$
$=\cot \left(\cot ^{-1} \frac{3}{4}\right)$
$=\frac{3}{4}$
(ix)
$\cos \left(\tan ^{-1} \frac{24}{7}\right)=\cos \left[\cos ^{-1} \frac{1}{\sqrt{1+\left(\frac{24}{7}\right)^{2}}}\right] \quad\left[\because \tan ^{-1} x=\cos ^{-1} \frac{1}{\sqrt{1+x^{2}}}\right]$
$=\cos \left[\cos ^{-1} \frac{1}{\sqrt{1+\frac{576}{49}}}\right]$
$=\cos \left[\cos ^{-1} \frac{1}{\frac{25}{7}}\right]$
$=\cos \left[\cos ^{-1} \frac{7}{25}\right]$
$=\frac{7}{25}$
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