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

Options

A. `(pi)/(2)`
B. `(pi)/(4)`
C. `pi`
D. `(3pi)/(4)`

Explanation

To solve the problem, we start with the given equation:

\[ \sin^{-1} x + \sin^{-1} y = \frac{\pi}{2} \]

### Step 1: Rewrite \(\sin^{-1} x\) and \(\sin^{-1} y\)
We know from trigonometric identities that:

\[ \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \]

This means we can express \(\sin^{-1} x\) as:

\[ \sin^{-1} x = \frac{\pi}{2} - \cos^{-1} x \]

Similarly, we can express \(\sin^{-1} y\) as:

\[ \sin^{-1} y = \frac{\pi}{2} - \cos^{-1} y \]

### Step 2: Substitute into the original equation
Now, substituting these expressions into the original equation, we have:

\[ \left(\frac{\pi}{2} - \cos^{-1} x\right) + \left(\frac{\pi}{2} - \cos^{-1} y\right) = \frac{\pi}{2} \]

### Step 3: Simplify the equation
Combining the terms gives us:

\[ \frac{\pi}{2} + \frac{\pi}{2} - \cos^{-1} x - \cos^{-1} y = \frac{\pi}{2} \]

This simplifies to:

\[ \pi - \cos^{-1} x - \cos^{-1} y = \frac{\pi}{2} \]

### Step 4: Isolate \(\cos^{-1} x + \cos^{-1} y\)
Now, we can isolate \(\cos^{-1} x + \cos^{-1} y\):

\[ \cos^{-1} x + \cos^{-1} y = \pi - \frac{\pi}{2} \]

### Step 5: Final calculation
This simplifies to:

\[ \cos^{-1} x + \cos^{-1} y = \frac{\pi}{2} \]

### Conclusion
Thus, we conclude that:

\[ \cos^{-1} x + \cos^{-1} y = \frac{\pi}{2} \]

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