^ -1 3-~ ^ -1 (-2) is equal to - Vidyadip

Question

$\tan ^{-1} \sqrt{3}-~\sec ^{-1}(-2)$ is equal to

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Answer

Let us take
$\tan ^{-1}(\sqrt{3})=x$ Then we get,
$\tan x=\sqrt{3}=\tan \frac{\pi}{3}$
$\Rightarrow ~x=\frac{\pi}3$
We know that range of the principle value branch of $\tan^{-1}$ is $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$
Therefore, $x = \tan ^{-1}(\sqrt{3})=\frac{\pi}{3}$
Let $\sec ^{-1}(-2)=y$ then we get,
$\text { sec } \mathrm{y}=-2=-\sec \frac{\pi}{3}=\sec \left(\pi-\frac{\pi}{3}\right)=\sec \left(\frac{2 \pi}{3}\right)$
We know that range of the principle value branch of $\sec^{-1}$ is $[0, \pi]$
Therefore, $\sec ^{-1}(-2)=\frac{2 \pi}{3}$
Hence, $\tan ^{-1} \sqrt{3}-~\sec ^{-1}(-2)$= $\frac{\pi}3$$-\frac{2 \pi}{3}= -\frac {\pi}3$

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