Question

Prove that
${\sin }^{-1}\frac{3}{5}+{\sin }^{-1}\frac{8}{17}={\cos }^{-1}\frac{36}{85}$

Answer 1

Consider the L.H.S.

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

We know that

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

Therefore,

$={\sin }^{-1}(\frac{3}{5}\sqrt{1-\frac{64}{289}}+\frac{8}{17}\sqrt{1-\frac{9}{25}})$

$={\sin }^{-1}(\frac{3}{5}\sqrt{\frac{225}{289}}+\frac{8}{17}\sqrt{\frac{16}{25}})$

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

$={\sin }^{-1}(\frac{45}{5\times 17}+\frac{32}{5\times 17})$

$={\sin }^{-1}\left(\frac{77}{85}\right)$

We know that

${\sin }^{-1}x={\cos }^{-1}\sqrt{1-x^2}$

Therefore,

$={\cos }^{-1}\sqrt{1-{\left(\frac{77}{85}\right)}^2}$

$={\cos }^{-1}\sqrt{\frac{1296}{{85}^2}}$

$={\cos }^{-1}\frac{36}{85}$

Hence, proved.