Question:
Find the principal value of each of the following :
$\tan ^{-1} \sqrt{3}-\cot ^{-1}(-\sqrt{3})^{3}$
Solution:
$\tan ^{-1} \sqrt{3}-\cot ^{-1}(-\sqrt{3})$
Putting the value of $\tan ^{-1} \sqrt{3}$ and using the formula
$\cot ^{-1}(-x)=\pi-\cot ^{-1} x$
$=\frac{\pi}{3}-\left(\pi-\cot ^{-1}(\sqrt{3})\right)$
Putting the value of $\cot ^{-1}(\sqrt{3})$
$=\frac{\pi}{3}-\left(\pi-\frac{\pi}{6}\right)$
$=\frac{\pi}{3}-\frac{5 \pi}{6}$
$=-\frac{3 \pi}{6}$
$=-\frac{\pi}{2}$
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