Write the value of ^ -1 (1 3 )- ^ -1 (-1 3 ) - Vidyadip

Question

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

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Answer

Given $\sin ^{-1}\left(\frac{1}{3}\right)-\cos ^{-1}\left(-\frac{1}{3}\right)$
We know that $\cos ^{-1}(-\theta)=\pi-\cos ^{-1} \theta$
$=\sin ^{-1}\left(\frac{1}{3}\right)-\left(\pi-\cos ^{-1}\left(\frac{1}{3}\right)\right)$
$=\sin ^{-1}\left(\frac{1}{3}\right)-\pi+\cos ^{-1}\left(\frac{1}{3}\right)$
$=\sin ^{-1}\left(\frac{1}{3}\right)+\cos ^{-1}\left(\frac{1}{3}\right)-\pi$
$=\frac{\pi}{2}-\pi$
$=-\frac{\pi}{2}$
Therefore we have,
$\sin ^{-1}\left(\frac{1}{3}\right)-\cos ^{-1}\left(-\frac{1}{3}\right)=-\frac{\pi}{2}$

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