The value of $\sin \left(\cos ^{-1}\left(-\frac{1}{3}\right)-\sin ^{-1}\left(\frac{1}{3}\right)\right)$ is

The value of $\sin \left(\cos ^{-1}\left(-\frac{1}{3}\right)-\sin ^{-1}\left(\frac{1}{3}\right)\right)$ is
  1. 1
  2. 3
  3. 2
  4. 4

Solution

$\begin{aligned} & \sin \left(\cos ^{-1}\left(-\frac{1}{3}\right)-\sin ^{-1}\left(\frac{1}{3}\right)\right) \\ & =\sin \left(\cos ^{-1}\left(-\frac{1}{3}\right)-\left[\frac{\pi}{2}-\cos ^{-1}\left(\frac{1}{3}\right)\right]\right) \\ & =\sin \left(\left[\pi-\cos ^{-1}\left(\frac{1}{3}\right)\right]-\frac{\pi}{2}+\cos ^{-1}\left(\frac{1}{3}\right)\right) \\ & =\sin \left(\frac{\pi}{2}\right) \\ & =1\end{aligned}$

Asked in: MHT CET 2024 (11 May Shift 2)

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