Evaluate each of the following:
Question:

Evaluate each of the following:

(i) $\cot ^{-1} \frac{1}{\sqrt{3}}-\operatorname{cosec}^{-1}(-2)+\sec ^{-1}\left(\frac{2}{\sqrt{3}}\right)$

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

(iii) $\operatorname{cosec}^{-1}\left(-\frac{2}{\sqrt{3}}\right)+2 \cot ^{-1}(-1)$

(iv) $\tan ^{-1}\left(-\frac{1}{\sqrt{3}}\right)+\cot ^{-1}\left(\frac{1}{\sqrt{3}}\right)+\tan ^{-1}\left(\sin \left(-\frac{\pi}{2}\right)\right)$

Solution:

(i)

$\cot ^{-1} \frac{1}{\sqrt{3}}-\operatorname{cosec}^{-1}(-2)+\sec ^{-1}\left(\frac{2}{\sqrt{3}}\right)=\cot ^{-1}\left(\cot \frac{\pi}{3}\right)-\operatorname{cosec}^{-1}\left[\operatorname{cosec}\left(-\frac{\pi}{6}\right)\right]+\sec ^{-1}\left(\sec \frac{\pi}{6}\right)$

$=\frac{\pi}{3}+\frac{\pi}{6}+\frac{\pi}{6}$

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

(ii)

$\cot ^{-1}\left\{2 \cos \left(\sin ^{-1} \frac{\sqrt{3}}{2}\right)\right\}=\cot ^{-1}\left\{2 \cos \left[\sin ^{-1}\left(\sin \frac{\pi}{3}\right)\right]\right\}$

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

$=\cot ^{-1}\left(2 \times \frac{1}{2}\right)$

$=\cot ^{-1}(1)$

$=\cot ^{-1}\left(\tan \frac{\pi}{4}\right)$

$=\frac{\pi}{4}$

(iii)

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

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

$=\frac{7 \pi}{6}$

(iv)

$\tan ^{-1}\left(-\frac{1}{\sqrt{3}}\right)+\cot ^{-1}\left(\frac{1}{\sqrt{3}}\right)+\tan ^{-1}\left(\sin \left(-\frac{\pi}{2}\right)\right)=\tan ^{-1}\left[\tan \left(-\frac{\pi}{6}\right)\right]+\cot ^{-1}\left(\cot \frac{\pi}{3}\right)+\tan ^{-1}(-1)$

$=\tan ^{-1}\left[\tan \left(-\frac{\pi}{6}\right)\right]+\cot ^{-1}\left(\cot \frac{\pi}{3}\right)+\tan ^{-1}\left[\tan \left(-\frac{\pi}{4}\right)\right]$

$=-\frac{\pi}{6}+\frac{\pi}{3}-\frac{\pi}{4}$

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

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