Question

\(\cos[\cot^{-1}(-\sqrt3)+\frac{\pi}{6}]=\)

• A. 0
• B. 1
• C. \(\frac{1}{\sqrt2}\)
• D. -1

Answer 1

The Correct Option is D

We need to evaluate the expression \( \cos \left[ \cot^{-1} \left( -\sqrt{3} \right) + \frac{\pi}{6} \right] \). \textbf{Step 1: Understanding} \( \cot^{-1}(-\sqrt{3}) \): The function \( \cot^{-1}(x) \) gives the angle whose cotangent is \( x \). We need to find the angle \( \theta \) such that: \[ \cot(\theta) = -\sqrt{3} \] The angle \( \theta \) where \( \cot(\theta) = -\sqrt{3} \) is \( \frac{5\pi}{6} \), because \( \cot \left( \frac{5\pi}{6} \right) = -\sqrt{3} \). Thus, \[ \cot^{-1}(-\sqrt{3}) = \frac{5\pi}{6} \] \textbf{Step 2: Substituting into the original expression:} Now, substitute this value into the original expression: \[ \cos \left( \frac{5\pi}{6} + \frac{\pi}{6} \right) \] \textbf{Step 3: Simplifying the angle:} To simplify, add the angles: \[ \frac{5\pi}{6} + \frac{\pi}{6} = \frac{6\pi}{6} = \pi \] \textbf{Step 4: Finding \( \cos(\pi) \):} From trigonometric values, we know that: \[ \cos(\pi) = -1 \] Thus, the correct answer is (D) -1. 

Answer 2

Let \(y = \cos[\cot^{-1}(-\sqrt3)+\frac{\pi}{6}]\).

First, we need to find the value of \(\cot^{-1}(-\sqrt3)\). Since the range of \(\cot^{-1}(x)\) is \((0, \pi)\), we look for an angle in this interval whose cotangent is \(-\sqrt{3}\).

We know that \(\cot(\frac{5\pi}{6}) = -\sqrt{3}\). Therefore, \(\cot^{-1}(-\sqrt3) = \frac{5\pi}{6}\).

Now, we substitute this value back into the expression:

\(y = \cos[\frac{5\pi}{6} + \frac{\pi}{6}] = \cos[\frac{6\pi}{6}] = \cos(\pi)\).

We know that \(\cos(\pi) = -1\).

Therefore, \(\cos[\cot^{-1}(-\sqrt3)+\frac{\pi}{6}] = -1\).