The number of solutions of the equation `sin^(-1)x=(sinx)^(-1)` is/are

Options

A. one
B. two
C. three
D. zero

Answer

To solve the equation \( \sin^{-1} x = (\sin x)^{-1} \), we will analyze both sides of the equation step by step.

### Step 1: Understand the functions involved
The left side, \( \sin^{-1} x \), is the inverse sine function, which is defined for \( x \) in the interval \([-1, 1]\) and gives values in the interval \([-\frac{\pi}{2}, \frac{\pi}{2}]\).

The right side, \( (\sin x)^{-1} \), is the reciprocal of the sine function. The sine function is defined for all real numbers, but it is important to note that \( \sin x \) can only take values between -1 and 1. Therefore, \( (\sin x)^{-1} \) is defined wherever \( \sin x \neq 0 \).

### Step 2: Set the domains
Since \( \sin^{-1} x \) is only defined for \( x \in [-1, 1] \), we will restrict our analysis to this interval.

### Step 3: Analyze the equation
We can rewrite the equation as:
\[ \sin^{-1} x = \frac{1}{\sin x} \]
This implies:
\[ x = \sin\left(\frac{1}{\sin x}\right) \]

### Step 4: Find the intersections
To find the number of solutions, we will analyze the behavior of both sides of the equation:
- The left side, \( y = \sin^{-1} x \), is an increasing function from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) as \( x \) goes from -1 to 1.
- The right side, \( y = \frac{1}{\sin x} \), has vertical asymptotes at \( x = n\pi \) for integers \( n \) where \( \sin x = 0 \).

### Step 5: Graphical analysis
To visualize the number of solutions, we can sketch the graphs of \( y = \sin^{-1} x \) and \( y = \frac{1}{\sin x} \):
- The graph of \( y = \sin^{-1} x \) will start at \( (-1, -\frac{\pi}{2}) \) and end at \( (1, \frac{\pi}{2}) \).
- The graph of \( y = \frac{1}{\sin x} \) will have peaks and troughs, and it will be undefined at points where \( \sin x = 0 \).

### Step 6: Check for intersections
We need to check for intersections within the interval \( x \in [-1, 1] \):
- At \( x = 0 \), \( \sin^{-1}(0) = 0 \) and \( \frac{1}{\sin(0)} \) is undefined.
- As \( x \) approaches 0 from the positive side, \( \sin^{-1} x \) approaches 0, while \( \frac{1}{\sin x} \) approaches infinity.
- As \( x \) approaches 1, \( \sin^{-1}(1) = \frac{\pi}{2} \) and \( \frac{1}{\sin(1)} \) is a finite value.

### Step 7: Conclusion
From the graphical analysis, we can conclude that there is one intersection point in the interval \( [-1, 1] \) where both functions meet.

Thus, the number of solutions of the equation \( \sin^{-1} x = (\sin x)^{-1} \) is **1**.