If `sin^(-1) x + sin^(-1) y = (2pi)/3", then " cos^(-1) x + cos^(-1) y `

Options

A. `(2pi)/3`
B. `pi/3`
C. `pi/6`
D. `pi`

Answer

To solve the problem, we need to find the value of \( \cos^{-1} x + \cos^{-1} y \) given that \( \sin^{-1} x + \sin^{-1} y = \frac{2\pi}{3} \).

### Step-by-Step Solution:

1. **Use the property of inverse trigonometric functions**:
We know that:
\[ \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \]
This means we can express \( \cos^{-1} x \) in terms of \( \sin^{-1} x \):
\[ \cos^{-1} x = \frac{\pi}{2} - \sin^{-1} x \]
Similarly, for \( y \):
\[ \cos^{-1} y = \frac{\pi}{2} - \sin^{-1} y \]

2. **Substitute the expressions into the equation**:
We can substitute these expressions into the equation we have:
\[ \cos^{-1} x + \cos^{-1} y = \left(\frac{\pi}{2} - \sin^{-1} x\right) + \left(\frac{\pi}{2} - \sin^{-1} y\right) \]
This simplifies to:
\[ \cos^{-1} x + \cos^{-1} y = \pi - (\sin^{-1} x + \sin^{-1} y) \]

3. **Substitute the given value**:
We know from the problem statement that:
\[ \sin^{-1} x + \sin^{-1} y = \frac{2\pi}{3} \]
Substitute this into the equation:
\[ \cos^{-1} x + \cos^{-1} y = \pi - \frac{2\pi}{3} \]

4. **Simplify the expression**:
Now, we simplify \( \pi - \frac{2\pi}{3} \):
\[ \pi = \frac{3\pi}{3} \]
Therefore:
\[ \cos^{-1} x + \cos^{-1} y = \frac{3\pi}{3} - \frac{2\pi}{3} = \frac{\pi}{3} \]

5. **Final answer**:
Thus, we find that:
\[ \cos^{-1} x + \cos^{-1} y = \frac{\pi}{3} \]

### Summary:
The value of \( \cos^{-1} x + \cos^{-1} y \) is \( \frac{\pi}{3} \).