`cos{cos^(- 1)((-sqrt(3))/2)+pi/6}`

Explanation

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

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

The value of \( \cos^{-1}(x) \) gives us an angle \( \theta \) such that \( \cos(\theta) = x \). We need to find \( \theta \) for \( x = -\frac{\sqrt{3}}{2} \).

The cosine function is negative in the second quadrant. The reference angle whose cosine is \( \frac{\sqrt{3}}{2} \) is \( \frac{\pi}{6} \). Therefore, in the second quadrant, we have:

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

So, we have:

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

### Step 2: Substitute back into the expression

Now we substitute this value back into the original expression:

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

### Step 3: Simplify the angle

Now we simplify the angle:

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

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

Now we need to find \( \cos(\pi) \):

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

### Final Answer

Thus, the final answer is:

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

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