Evaluate the following
`(i) sin ( pi/2 - sin^(-1) ( (-1)/2))`
(ii) ` sin(pi/2 - sin^(-1)(- sqrt3/2))`

Answer

To evaluate the given expressions, we will use some properties of inverse trigonometric functions and trigonometric identities. Let's solve each part step by step.

### Part (i): Evaluate `sin(π/2 - sin^(-1)(-1/2))`

1. **Use the identity for inverse sine**:
\[ \sin^{-1}(-x) = -\sin^{-1}(x) \]
Therefore, we can rewrite:
\[ \sin^{-1}(-1/2) = -\sin^{-1}(1/2) \]

2. **Evaluate \(\sin^{-1}(1/2)\)**:
The value of \(\sin^{-1}(1/2)\) is \(\pi/6\) because \(\sin(\pi/6) = 1/2\).

3. **Substitute back**:
\[ \sin^{-1}(-1/2) = -\pi/6 \]

4. **Substitute into the original expression**:
\[ \sin\left(\frac{\pi}{2} - \left(-\frac{\pi}{6}\right)\right) = \sin\left(\frac{\pi}{2} + \frac{\pi}{6}\right) \]

5. **Combine the angles**:
\[ \frac{\pi}{2} + \frac{\pi}{6} = \frac{3\pi}{6} + \frac{\pi}{6} = \frac{4\pi}{6} = \frac{2\pi}{3} \]

6. **Evaluate \(\sin(2\pi/3)\)**:
\(\sin(2\pi/3) = \sin(\pi - \pi/3) = \sin(\pi/3) = \frac{\sqrt{3}}{2}\).

Thus, the answer for part (i) is:
\[ \sin\left(\frac{\pi}{2} - \sin^{-1}\left(-\frac{1}{2}\right)\right) = \frac{\sqrt{3}}{2} \]

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### Part (ii): Evaluate `sin(π/2 - sin^(-1)(-sqrt(3)/2))`

1. **Use the identity for inverse sine**:
\[ \sin^{-1}(-x) = -\sin^{-1}(x) \]
Therefore, we can rewrite:
\[ \sin^{-1}(-\sqrt{3}/2) = -\sin^{-1}(\sqrt{3}/2) \]

2. **Evaluate \(\sin^{-1}(\sqrt{3}/2)\)**:
The value of \(\sin^{-1}(\sqrt{3}/2)\) is \(\pi/3\) because \(\sin(\pi/3) = \sqrt{3}/2\).

3. **Substitute back**:
\[ \sin^{-1}(-\sqrt{3}/2) = -\pi/3 \]

4. **Substitute into the original expression**:
\[ \sin\left(\frac{\pi}{2} - \left(-\frac{\pi}{3}\right)\right) = \sin\left(\frac{\pi}{2} + \frac{\pi}{3}\right) \]

5. **Combine the angles**:
\[ \frac{\pi}{2} + \frac{\pi}{3} = \frac{3\pi}{6} + \frac{2\pi}{6} = \frac{5\pi}{6} \]

6. **Evaluate \(\sin(5\pi/6)\)**:
\(\sin(5\pi/6) = \sin(\pi - \pi/6) = \sin(\pi/6) = \frac{1}{2}\).

Thus, the answer for part (ii) is:
\[ \sin\left(\frac{\pi}{2} - \sin^{-1}\left(-\sqrt{3}/2\right)\right) = \frac{1}{2} \]

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### Summary of Answers:
(i) \(\frac{\sqrt{3}}{2}\)
(ii) \(\frac{1}{2}\)

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