Question

The value of $\tan^{-1}(\sqrt{3}) - \cot^{-1}(-\sqrt{3})$ will be:

• A. $\pi$
• B. $-\frac{\pi}{2}$
• C. $0$
• D. $2\sqrt{3} \pi$

Hint:

For inverse trigonometric functions, remember the standard identities and adjust based on the quadrant of the angle.

Answer 1

The Correct Option is C

Step 1: Recall the identity for inverse trigonometric functions.
We know that: \[ \tan^{-1}(x) + \cot^{-1}(x) = \frac{\pi}{2}, \text{ for } x > 0. \] However, for negative values of $x$, the identity adjusts based on the quadrant.

Step 2: Simplify the expression.
Given expression is: \[ \tan^{-1}(\sqrt{3}) - \cot^{-1}(-\sqrt{3}). \] First, use the identity for inverse tangent and cotangent: \[ \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} \text{(since tan}(\frac{\pi}{3}) = \sqrt{3}), \] \[ \cot^{-1}(-\sqrt{3}) = \frac{\pi}{3} \text{(since cot}(\frac{\pi}{3}) = \frac{1}{\sqrt{3}} \text{, and negative value flips the sign)}. \]

Step 3: Conclusion.
Thus, the value is: \[ \frac{\pi}{3} - \frac{\pi}{3} = 0. \] So, the correct answer is (C) $0$.