Solve : ^ -1 (1)+ ^ -1 (-1 2 )+ ^ -1 (-1 2 ) - Vidyadip

Question

Solve : $\tan ^{-1}(1)+\cos ^{-1}\left(-\frac{1}{2}\right)+\sin ^{-1}\left(-\frac{1}{2}\right)$

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Answer

$y=\tan ^{-1}(1)+\cos ^{-1}\left(-\frac{1}{2}\right)+\sin ^{-1}\left(-\frac{1}{2}\right)$
Now put the value of principal :$
\begin{aligned}
\tan ^{-1}(1) & =\frac{\pi}{4}, \cos ^{-1}\left(-\frac{1}{2}\right)=\pi-\cos ^{-1}\left(\frac{1}{2}\right) \\
& =\pi-\frac{\pi}{3}=\frac{2 \pi}{3} \\
\sin ^{-1}\left(-\frac{1}{2}\right) & =-\sin ^{-}\left(\frac{1}{2}\right)=-\frac{\pi}{6}
\end{aligned}
$
Put these values in the equation (i) :
$\begin{aligned} y & =\frac{\pi}{4}+\frac{2 \pi}{3}-\frac{\pi}{6}=\frac{3 \pi+8 \pi-2 \pi}{12} \\ & =\frac{9 \pi}{12}=\frac{3 \pi}{4} \text {}\end{aligned}$

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