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

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

Step 1: Evaluate the Inverse Cotangent Function:
First, determine the value of \( \cot^{-1}(-\sqrt{3}) \).
Let \( \theta = \cot^{-1}(-\sqrt{3}) \). This implies \( \cot \theta = -\sqrt{3} \).
Since \( \cot \left( \frac{2\pi}{3} \right) = -\sqrt{3} \), it follows that:
\( \theta = \frac{2\pi}{3} \).

Step 2: Substitute into the Original Expression:
Replace the inverse cotangent term with its evaluated value:

\( \cos \left( \frac{\pi}{6} + \frac{2\pi}{3} \right) \).
To add the fractions, find a common denominator:
\( \frac{\pi}{6} + \frac{4\pi}{6} = \frac{5\pi}{6} \).
Thus, the expression becomes \( \cos \left( \frac{5\pi}{6} \right) \).

Step 3: Calculate the Cosine Value:
Evaluate \( \cos \left( \frac{5\pi}{6} \right) \).
We know that \( \cos \left( \frac{5\pi}{6} \right) = -\cos \left( \frac{\pi}{6} \right) \).
Therefore, \( \cos \left( \frac{5\pi}{6} \right) = -\frac{\sqrt{3}}{2} \).

Conclusion:
The computed value of \( \cos \left( \frac{\pi}{6} + \cot^{-1}(-\sqrt{3}) \right) \) is \( \boxed{ -\frac{\sqrt{3}}{2} } \).