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

To solve the problem, we need to evaluate the expression:
\( \cos \left( \frac{\pi}{6} + \cot^{-1}(-\sqrt{3}) \right) \)

1. Understand the Inverse Cotangent:
We start by evaluating \( \cot^{-1}(-\sqrt{3}) \).
Let \( \theta = \cot^{-1}(-\sqrt{3}) \)
Then, \( \cot \theta = -\sqrt{3} \)
We know that \( \cot \left( \frac{2\pi}{3} \right) = -\sqrt{3} \), so:
\( \theta = \cot^{-1}(-\sqrt{3}) = \frac{2\pi}{3} \)

2. Plug Back into the Original Expression:
Now substitute this value into the original expression:

\( \cos \left( \frac{\pi}{6} + \frac{2\pi}{3} \right) = \cos \left( \frac{\pi}{6} + \frac{4\pi}{6} \right) = \cos \left( \frac{5\pi}{6} \right) \)

3. Evaluate the Cosine:
\( \cos \left( \frac{5\pi}{6} \right) = -\cos \left( \frac{\pi}{6} \right) = -\frac{\sqrt{3}}{2} \)

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