Question

Find the value of $ \cot^{-1}(1) + \cot^{-1}(2) + \cot^{-1}(3) $

• A. \( \pi \)
• B. \( \frac{\pi}{2} \)
• C. \( \frac{\pi}{4} \)
• D. \( 2\pi \)

Hint:

Use identities for cotangents and their sums to simplify inverse trigonometric expressions like this one.

Answer 1

The Correct Option is B

We are given: \[ \cot^{-1}(1) + \cot^{-1}(2) + \cot^{-1}(3) \] Using the identity for the sum of inverse cotangents: \[ \cot^{-1}(x) + \cot^{-1}(y) = \cot^{-1} \left( \frac{xy - 1}{x + y} \right) \] First, compute \( \cot^{-1}(1) + \cot^{-1}(2) \): \[ \cot^{-1}(1) + \cot^{-1}(2) = \cot^{-1} \left( \frac{1 \times 2 - 1}{1 + 2} \right) = \cot^{-1} \left( \frac{1}{3} \right) \] Now, add \( \cot^{-1}(3) \): \[ \cot^{-1} \left( \frac{1}{3} \right) + \cot^{-1}(3) = \cot^{-1} \left( \frac{\left( \frac{1}{3} \right) \times 3 - 1}{\frac{1}{3} + 3} \right) = \cot^{-1}(0) \] Thus: \[ \cot^{-1}(0) = \frac{\pi}{2} \] So the answer is \( \frac{\pi}{2}\).