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

Options

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

Explanation

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 1: Use the identity for inverse sine and cosine
We know that:
\[ \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \]
and
\[ \sin^{-1} y + \cos^{-1} y = \frac{\pi}{2} \]

### Step 2: Rearranging the identities
From the above identities, we can express \( \cos^{-1} x \) and \( \cos^{-1} y \) in terms of \( \sin^{-1} x \) and \( \sin^{-1} y \):
\[ \cos^{-1} x = \frac{\pi}{2} - \sin^{-1} x \]
\[ \cos^{-1} y = \frac{\pi}{2} - \sin^{-1} y \]

### Step 3: Substitute into the expression
Now, substituting these into the expression \( \cos^{-1} x + \cos^{-1} y \):
\[ \cos^{-1} x + \cos^{-1} y = \left(\frac{\pi}{2} - \sin^{-1} x\right) + \left(\frac{\pi}{2} - \sin^{-1} y\right) \]
\[ = \pi - (\sin^{-1} x + \sin^{-1} y) \]

### Step 4: Substitute the given value
We know from the problem statement that \( \sin^{-1} x + \sin^{-1} y = \frac{2\pi}{3} \):
\[ \cos^{-1} x + \cos^{-1} y = \pi - \frac{2\pi}{3} \]

### Step 5: Simplify the expression
Now, simplifying the right side:
\[ \pi - \frac{2\pi}{3} = \frac{3\pi}{3} - \frac{2\pi}{3} = \frac{\pi}{3} \]

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

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