Find the value of : `cos[(pi)/3-"sin"^(-1)(-1/2)]`

Explanation

To solve the problem of finding the value of \( \cos\left(\frac{\pi}{3} - \sin^{-1}\left(-\frac{1}{2}\right)\right) \), we can follow these steps:

### Step 1: Find \( \sin^{-1}\left(-\frac{1}{2}\right) \)
The value of \( \sin^{-1}(x) \) gives us the angle whose sine is \( x \). We know that:
\[ \sin\left(-\frac{\pi}{6}\right) = -\frac{1}{2} \]
Thus, we have:
\[ \sin^{-1}\left(-\frac{1}{2}\right) = -\frac{\pi}{6} \]

### Step 2: Substitute into the cosine expression
Now we substitute this result back into our original expression:
\[ \cos\left(\frac{\pi}{3} - \left(-\frac{\pi}{6}\right)\right) = \cos\left(\frac{\pi}{3} + \frac{\pi}{6}\right) \]

### Step 3: Simplify the angle
Next, we need to simplify the angle:
\[ \frac{\pi}{3} + \frac{\pi}{6} = \frac{2\pi}{6} + \frac{\pi}{6} = \frac{3\pi}{6} = \frac{\pi}{2} \]

### Step 4: Calculate the cosine
Now we find the cosine of the simplified angle:
\[ \cos\left(\frac{\pi}{2}\right) = 0 \]

### Final Answer
Thus, the value of \( \cos\left(\frac{\pi}{3} - \sin^{-1}\left(-\frac{1}{2}\right)\right) \) is:
\[ \boxed{0} \]