Find the value of ^ -1 (-2)- ^ -1 (1 2 ). - Vidyadip

Question

Find the value of $\sec ^{-1}(-2)-\sin ^{-1}\left(\frac{1}{2}\right)$.

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Answer

$\sec ^{-1}(-2)-\sin ^{-1}\left(\frac{1}{2}\right)$
We know that :
$
\begin{aligned}
\sin ^{-1}\left(\frac{1}{2}\right) & =\frac{\pi}{6} \\
\cos \frac{2 \pi}{3} & =-\frac{1}{2} \\
\therefore \quad \sec \frac{2 \pi}{3} & =-2 \quad \therefore \sec ^{-1}(-2)=\frac{2 \pi}{3}
\end{aligned}
$
Putting the value
$\Rightarrow \quad \frac{2 \pi}{3}-\frac{\pi}{6}=\frac{4 \pi-\pi}{6}=\frac{3 \pi}{6}=\frac{\pi}{2}$

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