Mathematics - Inverse Trigonometric Functions Question with Solution | TestHub

${\sin }^{-1}\frac{4}{5}+{\sin }^{-1}\frac{5}{13}+{\sin }^{-1}\frac{16}{65}$

$={\tan }^{-1}\frac{4}{3}+{\tan }^{-1}\frac{5}{12}+{\tan }^{-1}\frac{16}{63}$

$={\tan }^{-1}\left(\frac{{\displaystyle \frac{4}{3}}+{\displaystyle \frac{5}{12}}}{1-{\displaystyle \frac{4}{3}}·{\displaystyle \frac{5}{12}}}\right)+{\tan }^{-1}\frac{16}{63}$

$={\tan }^{-1}\frac{63}{16}+{\tan }^{-1}\frac{16}{63}=\frac{\mathrm{\pi }}{2}$

$\therefore 2\mathrm{\pi }-\frac{\mathrm{\pi }}{2}=\frac{3\mathrm{\pi }}{2}$