Which of the following figure represents are correct solution of the equation `sin^(-1)xcos^(-1)y=0`?

Options

A.
B.
C.
D.

Answer

To solve the equation \( \sin^{-1}x \cos^{-1}y = 0 \), we can follow these steps:

### Step 1: Understand the equation
The equation \( \sin^{-1}x \cos^{-1}y = 0 \) implies that the product of two terms is zero. This means that at least one of the terms must be zero.

### Step 2: Set each term to zero
We can set each term in the product to zero:
1. \( \sin^{-1}x = 0 \)
2. \( \cos^{-1}y = 0 \)

### Step 3: Solve \( \sin^{-1}x = 0 \)
To solve \( \sin^{-1}x = 0 \):
- The value of \( x \) for which \( \sin^{-1}x = 0 \) is when \( x = \sin(0^\circ) \).
- Therefore, \( x = 0 \).

### Step 4: Solve \( \cos^{-1}y = 0 \)
To solve \( \cos^{-1}y = 0 \):
- The value of \( y \) for which \( \cos^{-1}y = 0 \) is when \( y = \cos(0^\circ) \).
- Therefore, \( y = 1 \).

### Step 5: Identify the solution set
From the above steps, we have:
- \( x = 0 \) (which corresponds to the y-axis)
- \( y = 1 \) (which corresponds to the horizontal line at \( y = 1 \))

### Step 6: Graph the solution
Now we can graph these solutions:
- The line \( x = 0 \) is the y-axis.
- The line \( y = 1 \) is a horizontal line that intersects the y-axis at \( y = 1 \).

### Conclusion
The correct representation of the solution to the equation \( \sin^{-1}x \cos^{-1}y = 0 \) is the intersection of the y-axis and the line \( y = 1 \).