If `sin((23pi)/(24))=sqrt((2sqrtp-sqrtq-1)/(4sqrtr))`, then the value of `(p^(2)+q^(2)-r^(2))` is equal to

Options

A. 6
B. 12
C. `-1`
D. 9

Answer

To solve the problem, we need to find the values of \( p \), \( q \), and \( r \) from the equation given, and then compute \( p^2 + q^2 - r^2 \).

### Step-by-Step Solution:

1. **Identify the given equation:**
\[ \sin\left(\frac{23\pi}{24}\right) = \sqrt{\frac{2\sqrt{p} - \sqrt{q} - 1}{4\sqrt{r}}} \]

2. **Use the sine identity:**
We can rewrite \( \sin\left(\frac{23\pi}{24}\right) \) using the identity \( \sin(\pi - x) = \sin(x) \):
\[ \sin\left(\frac{23\pi}{24}\right) = \sin\left(\pi - \frac{\pi}{24}\right) = \sin\left(\frac{\pi}{24}\right) \]

3. **Express \( \sin\left(\frac{\pi}{24}\right) \) using cosine:**
We can use the identity \( \sin^2\theta = \frac{1 - \cos(2\theta)}{2} \):
\[ \sin\left(\frac{\pi}{24}\right) = \sqrt{\frac{1 - \cos\left(\frac{\pi}{12}\right)}{2}} \]

4. **Calculate \( \cos\left(\frac{\pi}{12}\right) \):**
Rewrite \( \cos\left(\frac{\pi}{12}\right) \) using the cosine subtraction formula:
\[ \cos\left(\frac{\pi}{12}\right) = \cos\left(45^\circ - 30^\circ\right) = \cos(45^\circ)\cos(30^\circ) + \sin(45^\circ)\sin(30^\circ) \]
Substituting the known values:
\[ \cos(45^\circ) = \frac{1}{\sqrt{2}}, \quad \cos(30^\circ) = \frac{\sqrt{3}}{2}, \quad \sin(45^\circ) = \frac{1}{\sqrt{2}}, \quad \sin(30^\circ) = \frac{1}{2} \]
Thus,
\[ \cos\left(\frac{\pi}{12}\right) = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} + \frac{1}{\sqrt{2}} \cdot \frac{1}{2} = \frac{\sqrt{3} + 1}{2\sqrt{2}} \]

5. **Substitute back to find \( \sin\left(\frac{\pi}{24}\right) \):**
Now substituting \( \cos\left(\frac{\pi}{12}\right) \) back:
\[ \sin\left(\frac{\pi}{24}\right) = \sqrt{\frac{1 - \frac{\sqrt{3} + 1}{2\sqrt{2}}}{2}} = \sqrt{\frac{2\sqrt{2} - (\sqrt{3} + 1)}{4}} = \frac{\sqrt{2\sqrt{2} - \sqrt{3} - 1}}{2} \]

6. **Set the two expressions equal:**
Now we have:
\[ \frac{\sqrt{2\sqrt{2} - \sqrt{3} - 1}}{2} = \sqrt{\frac{2\sqrt{p} - \sqrt{q} - 1}{4\sqrt{r}}} \]

7. **Square both sides:**
Squaring both sides gives:
\[ \frac{2\sqrt{2} - \sqrt{3} - 1}{4} = \frac{2\sqrt{p} - \sqrt{q} - 1}{4\sqrt{r}} \]

8. **Cross-multiply to eliminate the fractions:**
\[ (2\sqrt{2} - \sqrt{3} - 1) \cdot 4\sqrt{r} = (2\sqrt{p} - \sqrt{q} - 1) \cdot 4 \]

9. **Compare coefficients to find \( p \), \( q \), and \( r \):**
From the comparison, we can deduce:
- \( p = 2 \)
- \( q = 3 \)
- \( r = 2 \)

10. **Calculate \( p^2 + q^2 - r^2 \):**
\[ p^2 + q^2 - r^2 = 2^2 + 3^2 - 2^2 = 4 + 9 - 4 = 9 \]

### Final Answer:
\[ p^2 + q^2 - r^2 = 9 \]