Write the principal value

Question:

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

Solution:

$\tan ^{-1} 1+\cos ^{-1}\left(-\frac{1}{2}\right)=\tan ^{-1}\left(\tan \frac{\pi}{4}\right)+\cos ^{-1}\left(\cos \frac{2 \pi}{3}\right)$

$=\frac{\pi}{4}+\frac{2 \pi}{3}$

$=\frac{11 \pi}{12}$

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