Question

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

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

Hint:

Use tan⁻¹(√3)=π/3 and cot⁻¹(-x)=π-cot⁻¹(x) with cot⁻¹(√3)=π/6.

Answer 1

The Correct Option is B

Step 1: Key Formula or Approach:
Use the standard values \(\tan^{-1}(\sqrt3) = \dfrac{\pi}{3}\) (range of \(\tan^{-1}\) is \((-\pi/2,\pi/2)\)), and for \(\cot^{-1}\) (range \((0,\pi)\)) use \(\cot^{-1}(-x) = \pi - \cot^{-1}(x)\).

Step 2: Evaluating each term:
\(\tan^{-1}(\sqrt3) = \dfrac{\pi}{3}\).
\(\cot^{-1}(\sqrt3) = \dfrac{\pi}{6}\), so \(\cot^{-1}(-\sqrt3) = \pi - \dfrac{\pi}{6} = \dfrac{5\pi}{6}\).

Step 3: Subtracting:
\[ \tan^{-1}(\sqrt3) - \cot^{-1}(-\sqrt3) = \frac{\pi}{3} - \frac{5\pi}{6} = \frac{2\pi - 5\pi}{6} = \frac{-3\pi}{6} = \frac{-\pi}{2} \]

Final Answer:
The value is \(-\pi/2\). \[ \boxed{-\dfrac{\pi}{2}} \]