Write the value

Question:

Write the value of $2 \sin ^{-1} \frac{1}{2}+\cos ^{-1}\left(-\frac{1}{2}\right)$.

Solution:

$2 \sin ^{-1} \frac{1}{2}+\cos ^{-1}\left(-\frac{1}{2}\right)=\sin ^{-1} 2 \times \frac{1}{2} \sqrt{1-\left(\frac{1}{2}\right)^{2}}+\cos ^{-1}\left(-\frac{1}{2}\right)$

$=\sin ^{-1} \frac{\sqrt{3}}{2}+\cos ^{-1}\left(-\frac{1}{2}\right)$

$=\sin ^{-1}\left(\sin \frac{\pi}{3}\right)+\cos ^{-1}\left(\cos \frac{2 \pi}{3}\right)$

$=\frac{\pi}{3}+\frac{2 \pi}{3}$

$=\pi$

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