Find the value of
`cos [cos^(-1) ((-sqrt3)/2) + pi/6]`

Answer

To find the value of \( \cos \left[ \cos^{-1} \left( -\frac{\sqrt{3}}{2} \right) + \frac{\pi}{6} \right] \), we can follow these steps:

### Step 1: Simplify \( \cos^{-1} \left( -\frac{\sqrt{3}}{2} \right) \)
We know that \( \cos^{-1} \left( -\frac{\sqrt{3}}{2} \right) \) corresponds to an angle in the second quadrant where the cosine value is negative. The angle that satisfies this condition is:
\[ \cos^{-1} \left( -\frac{\sqrt{3}}{2} \right) = \frac{5\pi}{6} \]

### Step 2: Substitute the angle back into the expression
Now we can substitute this value back into our expression:
\[ \cos \left[ \frac{5\pi}{6} + \frac{\pi}{6} \right] \]

### Step 3: Combine the angles
Adding the angles gives us:
\[ \frac{5\pi}{6} + \frac{\pi}{6} = \frac{6\pi}{6} = \pi \]

### Step 4: Calculate the cosine of the resulting angle
Now we need to find:
\[ \cos(\pi) \]
The cosine of \( \pi \) is:
\[ \cos(\pi) = -1 \]

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