The value of $\frac{ an ^{-1}(\sqrt{3})-\sec ^{-1}(-2)}{\operatorname{cosec}^{-1}(-\sqrt{2})+\cos ^{-1}\left(\frac{-1}{2} ight)}$ is

Question

The value of $\frac{	an ^{-1}(\sqrt{3})-\sec ^{-1}(-2)}{\operatorname{cosec}^{-1}(-\sqrt{2})+\cos ^{-1}\left(\frac{-1}{2}ight)}$ is

Aicharya · Accepted Answer

$\begin{aligned} & \frac{	an ^{-1}(\sqrt{3})-\sec ^{-1}(-2)}{\operatorname{cosec}^{-1}(-\sqrt{2})+\cos ^{-1}\left(\frac{-1}{2}ight)} \ & =\frac{	an ^{-1}(\sqrt{3})-\cos ^{-1}\left(\frac{-1}{2}ight)}{\sin ^{-1}\left(\frac{-1}{\sqrt{2}}ight)+\cos ^{-1}\left(\frac{-1}{2}ight)}\end{aligned}$ $\cdots\left[\begin{array}{l}\sec ^{-1} x=\cos ^{-1} \frac{1}{x} \ \operatorname{cosec}^{-1} x=\sin ^{-1} \frac{1}{x}\end{array}ight]$ $=\frac{	an ^{-1}(\sqrt{3})-\pi+\cos ^{-1}\left(\frac{1}{2}ight)}{-\sin ^{-1}\left(\frac{1}{\sqrt{2}}ight)+\pi-\cos ^{-1}\left(\frac{1}{2}ight)}$ $\ldots\left[\begin{array}{l}\cos ^{-1}(-x)=\pi-\cos ^{-1} x \ \sin ^{-1}(-x)=-\sin ^{-1} x\end{array}ight]$ $\begin{aligned} & =\frac{\frac{\pi}{3}-\pi+\frac{\pi}{3}}{-\frac{\pi}{4}+\pi-\frac{\pi}{3}} \ & =\frac{-\frac{\pi}{3}}{\frac{5 \pi}{12}} \ & =\frac{-4}{5}\end{aligned}$

The value of $\frac{\tan ^{-1}(\sqrt{3})-\sec ^{-1}(-2)}{\operatorname{cosec}^{-1}(-\sqrt{2})+\cos…

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