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

Explanation

To solve the equation \( \sin^{-1} x + \sin^{-1} y = \frac{2\pi}{3} \) and find the value of \( \cos^{-1} x + \cos^{-1} y \), we can follow these steps:

### Step 1: Write the given equation
We start with the equation:
\[ \sin^{-1} x + \sin^{-1} y = \frac{2\pi}{3} \]

### Step 2: Convert \( \sin^{-1} \) to \( \cos^{-1} \)
Using the identity \( \sin^{-1} z = \frac{\pi}{2} - \cos^{-1} z \), we can rewrite the equation:
\[ \left(\frac{\pi}{2} - \cos^{-1} x\right) + \left(\frac{\pi}{2} - \cos^{-1} y\right) = \frac{2\pi}{3} \]

### Step 3: Simplify the equation
Now, simplify the left-hand side:
\[ \frac{\pi}{2} + \frac{\pi}{2} - \cos^{-1} x - \cos^{-1} y = \frac{2\pi}{3} \]
This simplifies to:
\[ \pi - \cos^{-1} x - \cos^{-1} y = \frac{2\pi}{3} \]

### Step 4: Rearrange the equation
Now, we can rearrange the equation to isolate \( \cos^{-1} x + \cos^{-1} y \):
\[ -\cos^{-1} x - \cos^{-1} y = \frac{2\pi}{3} - \pi \]
This simplifies to:
\[ -\cos^{-1} x - \cos^{-1} y = \frac{2\pi}{3} - \frac{3\pi}{3} \]
\[ -\cos^{-1} x - \cos^{-1} y = -\frac{\pi}{3} \]

### Step 5: Multiply by -1
Multiplying both sides by -1 gives:
\[ \cos^{-1} x + \cos^{-1} y = \frac{\pi}{3} \]

### Final Result
Thus, the value of \( \cos^{-1} x + \cos^{-1} y \) is:
\[ \cos^{-1} x + \cos^{-1} y = \frac{\pi}{3} \]

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