Question

Prove: $si{n}^{-}1\frac{3}{5}+si{n}^{-}1\frac{8}{17}=si{n}^{-}1\frac{77}{85}$

Answer 1

${\sin }^{-1}\frac{3}{5}+{\sin }^{-1}\frac{8}{17}$

${\sin }^{-1}x+{\sin }^{-1}y$

${\sin }^{-1}(x\sqrt{1-y^2}+y\sqrt{1-x^2})$

$={\sin }^{-1}[\frac{3}{5}\sqrt{1-{\left(\frac{8}{17}\right)}^2}+\frac{8}{17}\sqrt{1-{\left(\frac{3}{5}\right)}^2}]$

${\sin }^{-1}[\frac{3}{5}\times \sqrt{\frac{225}{289}}+\frac{8}{17}\sqrt{16}27]$

${\sin }^{-1}[\frac{3}{5}\times \frac{15}{17}+\frac{8}{17}\times \frac{4}{5}]$

${\sin }^{-1}\left[\frac{45+32}{85}\right]$

$={\sin }^{-1}\left[\frac{77}{85}\right]=$RHS.