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

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