Evaluate `sin{pi/2-sin^(-1)(-sqrt(3)/2)}`.

Explanation

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

### Step 1: Rewrite the expression using properties of inverse trigonometric functions
We know that:
\[ \sin^{-1}(-x) = -\sin^{-1}(x) \]
Thus, we can rewrite \( \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) \) as:
\[ -\sin^{-1}\left(\frac{\sqrt{3}}{2}\right) \]
So, the expression becomes:
\[ \sin\left(\frac{\pi}{2} - \left(-\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)\right)\right) = \sin\left(\frac{\pi}{2} + \sin^{-1}\left(\frac{\sqrt{3}}{2}\right)\right) \]

### Step 2: Find \( \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) \)
The value of \( \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) \) corresponds to an angle where the sine is \( \frac{\sqrt{3}}{2} \). This angle is:
\[ \frac{\pi}{3} \]
Thus, we can substitute this back into our expression:
\[ \sin\left(\frac{\pi}{2} + \frac{\pi}{3}\right) \]

### Step 3: Simplify the angle
Now, we need to simplify the angle:
\[ \frac{\pi}{2} + \frac{\pi}{3} = \frac{3\pi}{6} + \frac{2\pi}{6} = \frac{5\pi}{6} \]
So now our expression is:
\[ \sin\left(\frac{5\pi}{6}\right) \]

### Step 4: Evaluate \( \sin\left(\frac{5\pi}{6}\right) \)
The angle \( \frac{5\pi}{6} \) is in the second quadrant where sine is positive. The sine of \( \frac{5\pi}{6} \) can be calculated as:
\[ \sin\left(\frac{5\pi}{6}\right) = \sin\left(\pi - \frac{\pi}{6}\right) = \sin\left(\frac{\pi}{6}\right) \]
We know that:
\[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]

### Final Answer
Thus, the value of the original expression is:
\[ \sin\left(\frac{\pi}{2} - \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right) = \frac{1}{2} \]