\displaystyle \textbf{Question 1: }\text{Find the principal value of each of the following:}
\displaystyle \text{(i) }\tan^{-1}\left(\frac{1}{\sqrt3}\right)\qquad \text{(ii) }\tan^{-1}\left(-\frac{1}{\sqrt3}\right)
\displaystyle \text{(iii) }\tan^{-1}\left(\cos\frac{\pi}{2}\right)\qquad \text{(iv) }\tan^{-1}\left(2\cos\frac{2\pi}{3}\right)
\displaystyle \text{Answer:}
\displaystyle \text{For any }x\in\mathbb{R},\ \tan^{-1}x\text{ represents an angle in }\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\text{ whose tangent is }x.
\displaystyle \text{(i) }\tan^{-1}\left(\frac{1}{\sqrt3}\right)=\tan^{-1}\left(\tan\frac{\pi}{6}\right)=\frac{\pi}{6}.
\displaystyle \text{(ii) }\tan^{-1}\left(-\frac{1}{\sqrt3}\right)=\tan^{-1}\left(\tan\left(-\frac{\pi}{6}\right)\right)=-\frac{\pi}{6}.
\displaystyle \text{(iii) }\tan^{-1}\left(\cos\frac{\pi}{2}\right)=\tan^{-1}(0)
\displaystyle =\tan^{-1}(\tan0)=0.
\displaystyle \text{(iv) }\tan^{-1}\left(2\cos\frac{2\pi}{3}\right)=\tan^{-1}(-1)
\displaystyle =\tan^{-1}\left(\tan\left(-\frac{\pi}{4}\right)\right)=-\frac{\pi}{4}.
\displaystyle \\

\displaystyle \textbf{Question 2: }\text{Find the principal values and evaluate each of the following:}
\displaystyle \text{(i) }\tan^{-1}(-1)+\cos^{-1}\left(-\frac{1}{\sqrt2}\right)
\displaystyle \text{(ii) }\tan^{-1}\left\{2\sin\left(4\cos^{-1}\frac{\sqrt3}{2}\right)\right\}
\displaystyle \text{Answer:}
\displaystyle \text{(i) }\tan^{-1}(-1)+\cos^{-1}\left(-\frac{1}{\sqrt2}\right)
\displaystyle =\tan^{-1}\left(\tan\left(-\frac{\pi}{4}\right)\right)+\cos^{-1}\left(\cos\frac{3\pi}{4}\right)
\displaystyle =-\frac{\pi}{4}+\frac{3\pi}{4}
\displaystyle =\frac{\pi}{2}
\displaystyle \text{(ii) }\tan^{-1}\left\{2\sin\left(4\cos^{-1}\frac{\sqrt3}{2}\right)\right\}
\displaystyle =\tan^{-1}\left\{2\sin\left(4\cdot\frac{\pi}{6}\right)\right\}
\displaystyle =\tan^{-1}\left(2\sin\frac{2\pi}{3}\right)
\displaystyle =\tan^{-1}\left(2\cdot\frac{\sqrt3}{2}\right)
\displaystyle =\tan^{-1}(\sqrt3)
\displaystyle =\tan^{-1}\left(\tan\frac{\pi}{3}\right)
\displaystyle =\frac{\pi}{3}
\displaystyle \\

\displaystyle \textbf{Question 3: }\text{Evaluate each of the following:}
\displaystyle \text{(i) }\tan^{-1}(-1)+\cos^{-1}\left(-\frac12\right)+\sin^{-1}\left(-\frac12\right)
\displaystyle \text{(ii) }\tan^{-1}\left(-\frac1{\sqrt3}\right)+\tan^{-1}(-\sqrt3)+\tan^{-1}\left(\sin\left(-\frac{\pi}{2}\right)\right)
\displaystyle \text{(iii) }\tan^{-1}\left(\tan\frac{5\pi}{6}\right)+\cos^{-1}\left(\cos\frac{13\pi}{6}\right)
\displaystyle \text{Answer:}

\displaystyle \text{(i) }\tan^{-1}(-1)+\cos^{-1}\left(-\frac12\right)+\sin^{-1}\left(-\frac12\right)
\displaystyle =\tan^{-1}\left(\tan\left(-\frac{\pi}{4}\right)\right)+\cos^{-1}\left(\cos\frac{2\pi}{3}\right)+\sin^{-1}\left(\sin\left(-\frac{\pi}{6}\right)\right)
\displaystyle =-\frac{\pi}{4}+\frac{2\pi}{3}-\frac{\pi}{6}
\displaystyle =\frac{\pi}{4}

\displaystyle \text{(ii) }\tan^{-1}\left(-\frac1{\sqrt3}\right)+\tan^{-1}(-\sqrt3)+\tan^{-1}\left(\sin\left(-\frac{\pi}{2}\right)\right)
\displaystyle =-\tan^{-1}\left(\frac1{\sqrt3}\right)-\tan^{-1}(\sqrt3)+\tan^{-1}(-1)
\displaystyle =-\tan^{-1}\left(\tan\frac{\pi}{6}\right)-\tan^{-1}\left(\tan\frac{\pi}{3}\right)-\tan^{-1}\left(\tan\frac{\pi}{4}\right)
\displaystyle =-\frac{\pi}{6}-\frac{\pi}{3}-\frac{\pi}{4}
\displaystyle =-\frac{3\pi}{4}

\displaystyle \text{(iii) }\tan^{-1}\left(\tan\frac{5\pi}{6}\right)+\cos^{-1}\left(\cos\frac{13\pi}{6}\right)
\displaystyle =\tan^{-1}\left(-\tan\frac{\pi}{6}\right)+\cos^{-1}\left(\cos\frac{\pi}{6}\right)
\displaystyle =-\frac{\pi}{6}+\frac{\pi}{6}
\displaystyle =0
\displaystyle \\

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