Show that `sin^(-1)(3/5)-sin^(-1)(8/(17))=cos^(-1)((84)/(85))` .

Answer

Proved

Explanation

Let `\sin ^{-1}(3 / 5)=x` and `\sin ^{-1}(8 / 17)=y`


Therefore `\sin x=3 / 5` and siny `=8 / 17`


Now, `.\cos x=\sqrt{( 1-\sin ^{2} x)}=\sqrt{(1-(3 / 5)^{2})}=\sqrt{(1-9 / 25)}=4 / 5`


and `\cos y=\sqrt{(1-\sin ^{2} y)}=\sqrt{(1-(8 / 17)^{2})}=\sqrt{(1-64 / 289)}=15 / 17`


We have `\cos (x-y)=\cos x \cos y+\sin x \sin y`

$$
\begin{aligned}
&=4/5 \times 15/17 + 3/5 \times 8/17 = 60/85 + 24/85 = 84/85 \\
...