Question

The value of \( \cos \left( \frac{\pi}{6} + \cot^{-1}(-\sqrt{3}) \right) \) is:

• A. \( -1 \)
• B. \( \frac{-\sqrt{3}}{2} \)
• C. \( 0 \)
• D. \( 1 \)

Hint:

When solving inverse trigonometric functions, express the angle in terms of a known trigonometric identity and simplify the expression.

Answer 1

The Correct Option is B

Step 1: Evaluating the inverse cotangent.
We know that \( \cot^{-1}(-\sqrt{3}) \) corresponds to an angle \( \theta \) where \( \cot \theta = -\sqrt{3} \). This implies \( \theta = \frac{5\pi}{6} \), because \( \cot \frac{5\pi}{6} = -\sqrt{3} \). Step 2: Simplifying the expression.
Thus, the expression becomes: \[ \cos \left( \frac{\pi}{6} + \frac{5\pi}{6} \right) = \cos \pi = -1 \] Therefore, the value of the expression is \( \boxed{-1} \).