Question

Show that ${\sin }^{-1}\left(\frac{8}{17}\right)+{\sin }^{-1}\left(\frac{3}{5}\right)={\sin }^{-1}\left(\frac{77}{85}\right)$

Answer 1

L.H.S

${\sin }^{-1}\left(\frac{8}{17}\right)+{\sin }^{-1}\left(\frac{3}{5}\right)$

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

${\sin }^{-1}[\frac{9}{17}+\frac{32}{85}]$

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

$\Rightarrow {\sin }^{-1}\left[\frac{77}{85}\right]$

R.H.S

Hence, proved.