If `sin^(-1)x+sin^(-1)y=(2pi)/3` and `cos^(-1)x-cos^(-1)y=-pi/3` then the number of values of `(x,y)` is

Explanation

To solve the problem step by step, we start with the given equations:

1. **Given Equations**:
\[ \sin^{-1} x + \sin^{-1} y = \frac{2\pi}{3} \quad \text{(1)} \]
\[ \cos^{-1} x - \cos^{-1} y = -\frac{\pi}{3} \quad \text{(2)} \]

2. **Using the Identity**:
We know that:
\[ \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \]
and
\[ \sin^{-1} y + \cos^{-1} y = \frac{\pi}{2} \]

3. **Adding Equations (1) and (2)**:
Adding the two equations gives:
\[ \sin^{-1} x + \sin^{-1} y + \cos^{-1} x - \cos^{-1} y = \frac{2\pi}{3} - \frac{\pi}{3} \]
Simplifying the right side:
\[ \sin^{-1} x + \sin^{-1} y + \cos^{-1} x - \cos^{-1} y = \frac{\pi}{3} \]

4. **Substituting the Identities**:
Substitute the identities into the equation:
\[ \sin^{-1} x + \left(\frac{\pi}{2} - \sin^{-1} x\right) + \sin^{-1} y - \left(\frac{\pi}{2} - \sin^{-1} y\right) = \frac{\pi}{3} \]
This simplifies to:
\[ \sin^{-1} y - \sin^{-1} y + \frac{\pi}{2} - \frac{\pi}{2} = \frac{\pi}{3} \]
This does not yield new information, so we will proceed differently.

5. **Rearranging Equation (2)**:
Rearranging equation (2) gives:
\[ \cos^{-1} x = \cos^{-1} y - \frac{\pi}{3} \]

6. **Using the Cosine Identity**:
Taking cosine on both sides:
\[ x = \cos\left(\cos^{-1} y - \frac{\pi}{3}\right) \]
Using the cosine subtraction formula:
\[ x = \cos(\cos^{-1} y)\cos\left(\frac{\pi}{3}\right) + \sin(\cos^{-1} y)\sin\left(\frac{\pi}{3}\right) \]
Thus,
\[ x = y \cdot \frac{1}{2} + \sqrt{1 - y^2} \cdot \frac{\sqrt{3}}{2} \]
Simplifying gives:
\[ x = \frac{y}{2} + \frac{\sqrt{3}}{2}\sqrt{1 - y^2} \]

7. **Substituting into Equation (1)**:
Substitute \(x\) back into equation (1):
\[ \sin^{-1}\left(\frac{y}{2} + \frac{\sqrt{3}}{2}\sqrt{1 - y^2}\right) + \sin^{-1}(y) = \frac{2\pi}{3} \]

8. **Finding Values**:
Solving this equation will yield the values of \(y\) and subsequently \(x\). However, we can check the possible values of \(y\) within the range \([-1, 1]\).

9. **Conclusion**:
After solving, we find that:
\[ y = \frac{1}{2} \quad \text{and} \quad x = 1 \]
Thus, the only pair \((x, y)\) is \((1, \frac{1}{2})\).

10. **Final Answer**:
The number of values of \((x, y)\) is **1**.