Evaluate `cos [(pi)/(6) + cos^(-1) (-(sqrt(3))/(2))]`

Answer

To evaluate the expression \( \cos \left( \frac{\pi}{6} + \cos^{-1} \left(-\frac{\sqrt{3}}{2}\right) \right) \), we will follow these steps:

### Step 1: Simplify \( \cos^{-1} \left(-\frac{\sqrt{3}}{2}\right) \)

The value \( \cos^{-1} \left(-\frac{\sqrt{3}}{2}\right) \) corresponds to an angle where the cosine value is \(-\frac{\sqrt{3}}{2}\). This occurs at:

\[ \theta = \pi - \frac{\pi}{6} = \frac{5\pi}{6} \]

Thus, we can rewrite:

\[ \cos^{-1} \left(-\frac{\sqrt{3}}{2}\right) = \frac{5\pi}{6} \]

### Step 2: Substitute back into the original expression

Now we can substitute this back into our original expression:

\[ \cos \left( \frac{\pi}{6} + \frac{5\pi}{6} \right) \]

### Step 3: Combine the angles

Adding the angles together:

\[ \frac{\pi}{6} + \frac{5\pi}{6} = \frac{6\pi}{6} = \pi \]

### Step 4: Evaluate \( \cos(\pi) \)

Now we evaluate \( \cos(\pi) \):

\[ \cos(\pi) = -1 \]

### Final Answer

Thus, the value of \( \cos \left( \frac{\pi}{6} + \cos^{-1} \left(-\frac{\sqrt{3}}{2}\right) \right) \) is:

\[ \boxed{-1} \]

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