Class 12: Inverse Trigonometric Functions

$\displaystyle \textbf{Question 1: }\text{Evaluate:}$
$\displaystyle \text{(i) }\cos\left\{\sin^{-1}\left(-\frac{7}{25}\right)\right\}$
$\displaystyle \text{(ii) }\sec\left\{\cot^{-1}\left(-\frac{5}{12}\right)\right\}$
$\displaystyle \text{(iii) }\cot\left\{\sec^{-1}\left(-\frac{13}{5}\right)\right\}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{(i) }\cos\left\{\sin^{-1}\left(-\frac{7}{25}\right)\right\}$
$\displaystyle =\cos\left\{-\sin^{-1}\left(\frac{7}{25}\right)\right\}$
$\displaystyle =\cos\left\{\sin^{-1}\left(\frac{7}{25}\right)\right\}$
$\displaystyle =\cos\left\{\cos^{-1}\sqrt{1-\left(\frac{7}{25}\right)^2}\right\}$
$\displaystyle =\cos\left\{\cos^{-1}\frac{24}{25}\right\}$
$\displaystyle =\frac{24}{25}$
$\displaystyle \\$
$\displaystyle \text{(ii) }\sec\left\{\cot^{-1}\left(-\frac{5}{12}\right)\right\}$
$\displaystyle =\sec\left\{\pi-\cot^{-1}\left(\frac{5}{12}\right)\right\}$
$\displaystyle =-\sec\left\{\cot^{-1}\left(\frac{5}{12}\right)\right\}$
$\displaystyle =-\sec\left\{\cos^{-1}\frac{5}{13}\right\}$
$\displaystyle =-\sec\left\{\sec^{-1}\frac{13}{5}\right\}$
$\displaystyle =-\frac{13}{5}$
$\displaystyle \\$
$\displaystyle \text{(iii) }\cot\left\{\sec^{-1}\left(-\frac{13}{5}\right)\right\}$
$\displaystyle =\cot\left\{\pi-\sec^{-1}\left(\frac{13}{5}\right)\right\}$
$\displaystyle =-\cot\left\{\sec^{-1}\left(\frac{13}{5}\right)\right\}$
$\displaystyle =-\cot\left\{\tan^{-1}\frac{12}{5}\right\}$
$\displaystyle =-\cot\left\{\cot^{-1}\frac{5}{12}\right\}$
$\displaystyle =-\frac{5}{12}$
$\displaystyle \\$

$\displaystyle \textbf{Question 2: }\text{Evaluate:}$
$\displaystyle \text{(i) }\tan\left\{\cos^{-1}\left(-\frac{7}{25}\right)\right\}$
$\displaystyle \text{(ii) }\mathrm{cosec}\left\{\cot^{-1}\left(-\frac{12}{5}\right)\right\}$
$\displaystyle \text{(iii) }\cos\left\{\tan^{-1}\left(-\frac{3}{4}\right)\right\}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{(i) }\tan\left\{\cos^{-1}\left(-\frac{7}{25}\right)\right\}$
$\displaystyle =\tan\left\{\pi-\cos^{-1}\left(\frac{7}{25}\right)\right\}$
$\displaystyle =-\tan\left\{\cos^{-1}\left(\frac{7}{25}\right)\right\}$
$\displaystyle =-\tan\left\{\tan^{-1}\left(\frac{\sqrt{1-\left(\frac{7}{25}\right)^2}}{\frac{7}{25}}\right)\right\}$
$\displaystyle =-\tan\left\{\tan^{-1}\left(\frac{24}{7}\right)\right\}$
$\displaystyle =-\frac{24}{7}$
$\displaystyle \\$
$\displaystyle \text{(ii) }\mathrm{cosec}\left\{\cot^{-1}\left(-\frac{12}{5}\right)\right\}$
$\displaystyle =\mathrm{cosec}\left\{\pi-\cot^{-1}\left(\frac{12}{5}\right)\right\}$
$\displaystyle =\mathrm{cosec}\left\{\cot^{-1}\left(\frac{12}{5}\right)\right\}$
$\displaystyle =\mathrm{cosec}\left\{\sin^{-1}\left(\frac{1}{\sqrt{1+\left(\frac{12}{5}\right)^2}}\right)\right\}$
$\displaystyle =\mathrm{cosec}\left\{\sin^{-1}\left(\frac{5}{13}\right)\right\}$
$\displaystyle =\mathrm{cosec}\left\{\mathrm{cosec}^{-1}\left(\frac{13}{5}\right)\right\}$
$\displaystyle =\frac{13}{5}$
$\displaystyle \\$
$\displaystyle \text{(iii) }\cos\left\{\tan^{-1}\left(-\frac{3}{4}\right)\right\}$
$\displaystyle =\cos\left\{-\tan^{-1}\left(\frac{3}{4}\right)\right\}$
$\displaystyle =\cos\left\{\tan^{-1}\left(\frac{3}{4}\right)\right\}$
$\displaystyle =\cos\left\{\cos^{-1}\left(\frac{1}{\sqrt{1+\left(\frac{3}{4}\right)^2}}\right)\right\}$
$\displaystyle =\cos\left\{\cos^{-1}\left(\frac{1}{\sqrt{1+\frac{9}{16}}}\right)\right\}$
$\displaystyle =\cos\left\{\cos^{-1}\left(\frac{1}{\sqrt{\frac{25}{16}}}\right)\right\}$
$\displaystyle =\cos\left\{\cos^{-1}\left(\frac{4}{5}\right)\right\}$
$\displaystyle =\frac{4}{5}$
$\displaystyle \\$

$\displaystyle \textbf{Question 3: }\text{Evaluate: }\sin\left\{\cos^{-1}\left(-\frac35\right)+\cot^{-1}\left(-\frac{5}{12}\right)\right\}$
$\displaystyle \text{Answer:}$
$\displaystyle \sin\left\{\cos^{-1}\left(-\frac35\right)+\cot^{-1}\left(-\frac{5}{12}\right)\right\}$
$\displaystyle =\sin\left\{\pi-\cos^{-1}\left(\frac35\right)+\pi-\cot^{-1}\left(\frac{5}{12}\right)\right\}$
$\displaystyle =\sin\left\{2\pi-\left[\cos^{-1}\left(\frac35\right)+\cot^{-1}\left(\frac{5}{12}\right)\right]\right\}$
$\displaystyle =-\sin\left\{\cos^{-1}\left(\frac35\right)+\cot^{-1}\left(\frac{5}{12}\right)\right\}$
$\displaystyle =-\sin\left\{\sin^{-1}\frac45+\sin^{-1}\frac{12}{13}\right\}$
$\displaystyle =-\left[\sin\left(\sin^{-1}\frac45\right)\cos\left(\sin^{-1}\frac{12}{13}\right)+\cos\left(\sin^{-1}\frac45\right)\sin\left(\sin^{-1}\frac{12}{13}\right)\right]$
$\displaystyle =-\left[\frac45\times\sqrt{1-\left(\frac{12}{13}\right)^2}+\frac35\times\frac{12}{13}\right]$
$\displaystyle =-\left[\frac45\times\frac{5}{13}+\frac35\times\frac{12}{13}\right]$
$\displaystyle =-\left[\frac{20}{65}+\frac{36}{65}\right]$
$\displaystyle =-\frac{56}{65}$
$\displaystyle \\$