Question

(b) Find the value of \( \tan^{-1}(\sqrt{3}) - \cot^{-1}(-\sqrt{3}) \):

Hint:

Use trigonometric identities and inverse function properties for simplifications.

Answer 1

We know: \[ \tan^{-1}(\sqrt{3}) = \frac{\pi}{3}, \quad \cot^{-1}(-\sqrt{3}) = \tan^{-1}(-\frac{1}{\sqrt{3}}). \] Since \( \tan^{-1}(-x) = -\tan^{-1}(x) \): \[ \cot^{-1}(-\sqrt{3}) = -\tan^{-1}(\frac{1}{\sqrt{3}}) = -\frac{\pi}{6}. \] Thus: \[ \tan^{-1}(\sqrt{3}) - \cot^{-1}(-\sqrt{3}) = \frac{\pi}{3} - \left(-\frac{\pi}{6}\right) = \frac{\pi}{3} + \frac{\pi}{6} = \frac{\pi}{2}. \]