Question

\[ \begin{aligned} &\text { The value of }\\ &\begin{aligned} \sin ^{-1}\left(-\frac{1}{\sqrt{2}}\right)+\cos ^{-1} & \left(-\frac{1}{2}\right) -\cot ^{-1}\left(-\frac{1}{\sqrt{3}}\right)+\tan ^{-1}(-\sqrt{3}) \text { is } \end{aligned} \end{aligned} \]

• A. \(-\frac{7\pi}{12}\)
• B. \(\frac{\pi}{4}\)
• C. \(\frac{\pi}{3}\)
• D. \(\frac{\pi}{6}\)

Hint:

Use the principal value ranges of each inverse function and the identities for negative arguments.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
Each inverse function has a principal value range: \(\sin^{-1}\) in \([-\frac{\pi}{2},\frac{\pi}{2}]\), \(\cos^{-1}\) in \([0,\pi]\), \(\cot^{-1}\) in \((0,\pi)\) and \(\tan^{-1}\) in \((-\frac{\pi}{2},\frac{\pi}{2})\).

Step 2: Evaluate each term
\(\sin^{-1}\left(-\frac{1}{\sqrt{2}}\right)=-\frac{\pi}{4}\)
\(\cos^{-1}\left(-\frac{1}{2}\right)=\pi-\frac{\pi}{3}=\frac{2\pi}{3}\)
\(\cot^{-1}\left(-\frac{1}{\sqrt{3}}\right)=\pi-\frac{\pi}{3}=\frac{2\pi}{3}\)
\(\tan^{-1}(-\sqrt{3})=-\frac{\pi}{3}\)

Step 3: Add the terms
Sum \(=-\frac{\pi}{4}+\frac{2\pi}{3}-\frac{2\pi}{3}-\frac{\pi}{3}=-\frac{\pi}{4}-\frac{\pi}{3}=-\frac{7\pi}{12}\).

Step 4: Compare the options
The value \(-\frac{7\pi}{12}\) matches option (A).

Final Answer:
The value of the expression is \(-\frac{7\pi}{12}\). \[ \boxed{\text{(A) }-\dfrac{7\pi}{12}} \]