Question

Show that ${\sin }^{-1}\frac{3}{5}-{\sin }^{-1}\frac{8}{17}={\cos }^{-1}\frac{84}{85}$

Answer 1

Let ${\sin }^{-1}\frac{3}{5}=x\Rightarrow \sin x=\frac{3}{5}$
$\Rightarrow \cos x=\sqrt{1-{\sin }^{2}x}=\sqrt{1-\frac{9}{25}}=\sqrt{\frac{16}{25}}=\frac{4}{5}$

Let ${\sin }^{-1}\frac{8}{17}=y\Rightarrow \sin y=\frac{8}{17}$
$\Rightarrow \cos y=\sqrt{1-{\sin }^{2}y}=\sqrt{1-\frac{64}{289}}=\sqrt{\frac{225}{289}}=\frac{15}{17}$

Now, $\cos (x-y)=\cos x\cos y+\sin x\sin y$

$=\frac{4}{5}\times \frac{15}{17}+\frac{3}{5}\times \frac{8}{17}$

$=\frac{60+24}{17\times 5}=\frac{84}{85}$
$\therefore \cos (x-y)=\frac{84}{85}$
$\Rightarrow x-y={\cos }^{-1}\frac{84}{85}$
$\Rightarrow {\sin }^{-1}\frac{3}{5}-{\sin }^{-1}\frac{8}{17}={\cos }^{-1}\frac{84}{85}$