Question

Find \( \tan^{-1} (\sqrt{3}) - \cot^{-1} (-\sqrt{3}) \).

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

Hint:

When dealing with inverse trigonometric functions of negative values, use the formula \( \cot^{-1} x = \frac{\pi}{2} - \tan^{-1} x \) to simplify the expression.

Answer 1

The Correct Option is B

We are asked to find: \[ \tan^{-1} (\sqrt{3}) - \cot^{-1} (-\sqrt{3}) \] We know that: \[ \tan^{-1} (\sqrt{3}) = \frac{\pi}{3} \] Now, for \( \cot^{-1} (-\sqrt{3}) \), recall that: \[ \cot^{-1} x = \frac{\pi}{2} - \tan^{-1} x \] So, \[ \cot^{-1} (-\sqrt{3}) = \frac{\pi}{2} - \tan^{-1} (-\sqrt{3}) = \frac{\pi}{2} + \frac{\pi}{3} = \frac{5\pi}{6} \] Now, subtracting the two terms: \[ \tan^{-1} (\sqrt{3}) - \cot^{-1} (-\sqrt{3}) = \frac{\pi}{3} - \frac{5\pi}{6} = -\frac{\pi}{2} \] Thus, the correct answer is \( -\frac{\pi}{2} \).