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

Explanation

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

### Step 1: Simplify the expression
We start with the expression:
\[ \cos\left(\frac{\pi}{3} - \sin^{-1}\left(-\frac{1}{2}\right)\right) \]

### Step 2: Use the property of the inverse sine function
We know that:
\[ \sin^{-1}(-x) = -\sin^{-1}(x) \]
Thus, we can rewrite:
\[ \sin^{-1}\left(-\frac{1}{2}\right) = -\sin^{-1}\left(\frac{1}{2}\right) \]

### Step 3: Find \( \sin^{-1}\left(\frac{1}{2}\right) \)
The value of \( \sin^{-1}\left(\frac{1}{2}\right) \) corresponds to the angle whose sine is \( \frac{1}{2} \). This angle is:
\[ \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6} \]

### Step 4: Substitute back into the expression
Now substituting back, we have:
\[ \sin^{-1}\left(-\frac{1}{2}\right) = -\frac{\pi}{6} \]
Thus, our expression becomes:
\[ \cos\left(\frac{\pi}{3} - \left(-\frac{\pi}{6}\right)\right) = \cos\left(\frac{\pi}{3} + \frac{\pi}{6}\right) \]

### Step 5: Add the angles
To add the angles, we need a common denominator:
\[ \frac{\pi}{3} = \frac{2\pi}{6} \]
So,
\[ \frac{\pi}{3} + \frac{\pi}{6} = \frac{2\pi}{6} + \frac{1\pi}{6} = \frac{3\pi}{6} = \frac{\pi}{2} \]

### Step 6: Evaluate the cosine
Now we evaluate:
\[ \cos\left(\frac{\pi}{2}\right) \]
The value of \( \cos\left(\frac{\pi}{2}\right) \) is:
\[ 0 \]

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