Question

If $sin^{-1} x +sin^{-1} y = \frac{2\pi}{3} $ and $cos^{-1} x - cos^{-1}y = \frac{\pi}{3}$ Then, $(x, y)$ is equal to

• A. $(0, 1)$
• B. $(1/2, 1)$
• C. $(1, 1/2)$
• D. $( \sqrt 3/2, 1)$

Answer 1

Answer: (B)

We have,
$ \sin ^{-1} x+\sin ^{-1} y=\frac{2 \pi}{3} ....(i)$
$\left(\frac{\pi}{2}-\sin ^{-1} x\right)-\left(\frac{\pi}{2}-\sin ^{-1} y\right)=\frac{\pi}{3}$
$ \Rightarrow -\sin ^{-1} x+\sin ^{-1} y=\frac{\pi}{3} ...(ii)$
On adding Eqs. (i) and (ii),
$\sin ^{-1} y=\frac{\pi}{2} \Rightarrow y=1$
On subtracting Eq. (i) from Eq. (ii),
$\sin ^{-1} x=\frac{\pi}{6} \Rightarrow x=\frac{1}{2} $
$\therefore (x, y)=\left(\frac{1}{2}, 1\right)$