Question

Show that ${\cos }^{-1}\frac{3}{5}-{\cos }^{-1}\frac{8}{17}=-{\sin }^{-1}\frac{13}{85}$

Answer 1

$\text{L.H.S.}={\cos }^{-1}\frac{3}{5}-{\cos }^{-1}\frac{8}{17}$
$=\frac{\pi }{2}-{\sin }^{-1}\frac{3}{5}-(\frac{\pi }{2}-{\sin }^{-1}\frac{8}{17})$
$(\because {\sin }^{-1}x+{\cos }^{-1}x=\frac{\pi }{2})$

$={\sin }^{-1}\frac{8}{17}-{\sin }^{-1}\frac{3}{5}$
$={\sin }^{-1}(\frac{8}{17}\times \sqrt{1-{\left(\frac{3}{5}\right)}^2}-\frac{3}{5}\times \sqrt{1-{\left(\frac{8}{17}\right)}^2})$
$(\because {\sin }^{-1}x-{\sin }^{-1}y={\sin }^{-1}(x\sqrt{1-y^2}-y\sqrt{1-x^2}\left)\right)$

$={\sin }^{-1}(\frac{8}{17}\times \frac{4}{5}-\frac{15}{17}\times \frac{3}{5})$
$={\sin }^{-1}\frac{-13}{85}$
$=-{\sin }^{-1}\frac{13}{85}$

$\text{L.H.S.}=\text{R.H.S}$ (Proved)