Question

Prove that ${\sin }^{-1}\frac{4}{5}+{\sin }^{-1}\frac{5}{13}+{\sin }^{-1}\left(\frac{16}{65}\right)=\frac{\pi }{2}$

Answer 1

Given to prove ${\sin }^{-1}\left(\frac{4}{5}\right)+{\sin }^{-1}\left(\frac{5}{13}\right)+{\sin }^{-1}\left(\frac{16}{65}\right)=\frac{\pi }{2}$

$\Rightarrow {\sin }^{-1}\left(\frac{4}{5}\right)+{\sin }^{-1}\left(\frac{5}{13}\right)+{\sin }^{-1}\left(\frac{16}{65}\right)={\sin }^{-1}\left(1\right)$

$\Rightarrow {\sin }^{-1}\left(\frac{4}{5}\right)+{\sin }^{-1}\left(\frac{5}{13}\right)={\sin }^{-1}\left(1\right)-{\sin }^{-1}\left(\frac{16}{65}\right)$

Using formula

$LHS:{\sin }^{-1}\left(A\right)+{\sin }^{-1}\left(B\right)={\sin }^{-1}(A\sqrt{1-B^2}+B\sqrt{1-A^2})$

$\Rightarrow {\sin }^{-1}(\frac{4}{5}\sqrt{1-\frac{25}{169}}+\frac{5}{13}\sqrt{1-\frac{16}{25}})$

$\Rightarrow {\sin }^{-1}(\frac{4}{5}\times \sqrt{\frac{144}{169}}+\frac{5}{13}\sqrt{\frac{9}{25}})$

$={\sin }^{-1}(\frac{4}{5}\times \frac{12}{13}+\frac{5}{13}\times \frac{3}{5})$

$\Rightarrow {\sin }^{-1}(\frac{48}{65}+\frac{3}{13})$

$\Rightarrow {\sin }^{-1}\left(\frac{63}{65}\right)$

$RHS:{\sin }^{-1}1-{\sin }^{-1}\frac{16}{65}$

$\Rightarrow {\sin }^{-1}(1\sqrt{1-{\left(\frac{16}{65}\right)}^2}-\frac{16}{65}\sqrt{1-1^2})$

$\Rightarrow {\sin }^{-1}\left(\frac{63}{65}\right)$

$LHS=RHS$