Question

Evaluate: $ \cot^{-1} \left( -\frac{3}{\sqrt{3}} \right) - \sec^{-1} \left( -\frac{2}{\sqrt{2}} \right) - \csc^{-1}(-1) - \tan^{-1}(1) $

• A. \( \frac{\pi}{6} \)
• B. \( -\frac{2\pi}{3} \)
• C. \( 0 \)
• D. \( \frac{\pi}{3} \)

Hint:

When dealing with inverse trigonometric functions, evaluate each term separately and use standard angle values for known trigonometric ratios.

Answer 1

The Correct Option is D

We will evaluate each term separately. \[ \cot^{-1} \left( -\frac{3}{\sqrt{3}} \right) = \cot^{-1} \left( -\sqrt{3} \right) \] We know that \( \cot^{-1} \left( -\sqrt{3} \right) \) corresponds to an angle of \( \frac{2\pi}{3} \). Next, \[ \sec^{-1} \left( -\frac{2}{\sqrt{2}} \right) = \sec^{-1} \left( -\sqrt{2} \right) \] Since \( \sec^{-1} \left( -\sqrt{2} \right) \) corresponds to an angle of \( \frac{3\pi}{4} \). Next, \[ \csc^{-1}(-1) = \frac{3\pi}{2} \] because the cosecant function is \( -1 \) at this angle. Finally, \[ \tan^{-1}(1) = \frac{\pi}{4} \] since the tangent of \( 45^\circ \) or \( \frac{\pi}{4} \) is 1. Now, putting everything together: \[ \frac{2\pi}{3} - \frac{3\pi}{4} - \frac{3\pi}{2} - \frac{\pi}{4} \] We simplify and get: \[ \frac{2\pi}{3} - \frac{3\pi}{4} - \frac{3\pi}{2} - \frac{\pi}{4} = \frac{\pi}{3} \] Thus, the correct answer is \( \frac{\pi}{3} \).