If `"cosec" ^(-1) x = sin ^(-1) ((1)/(x))` then x may be -

Options

A. 1
B. `(-1)/(2)`
C. `(3)/(2) `
D. `(-3)/(2) `

Answer

To solve the equation \( \csc^{-1}(x) = \sin^{-1}\left(\frac{1}{x}\right) \), we need to analyze the conditions under which this equality holds true.

### Step 1: Understand the definitions
The cosecant function is the reciprocal of the sine function. Therefore, we can express the inverse cosecant function as:
\[ \csc^{-1}(x) = \theta \implies \csc(\theta) = x \implies \sin(\theta) = \frac{1}{x} \]
This means that the equation \( \csc^{-1}(x) = \sin^{-1}\left(\frac{1}{x}\right) \) is valid when both sides are defined.

### Step 2: Determine the domain of \( \csc^{-1}(x) \)
The function \( \csc^{-1}(x) \) is defined for:
\[ x \leq -1 \quad \text{or} \quad x \geq 1 \]
This means that \( x \) must be either less than or equal to -1 or greater than or equal to 1.

### Step 3: Determine the domain of \( \sin^{-1}\left(\frac{1}{x}\right) \)
The function \( \sin^{-1}(y) \) is defined for \( y \) in the range \([-1, 1]\). Therefore, for \( \sin^{-1}\left(\frac{1}{x}\right) \) to be defined, we need:
\[ -1 \leq \frac{1}{x} \leq 1 \]
This gives us two inequalities to solve.

1. From \( \frac{1}{x} \geq -1 \):
\[ 1 \geq -x \implies x \geq -1 \]

2. From \( \frac{1}{x} \leq 1 \):
\[ 1 \leq x \implies x \geq 1 \]

### Step 4: Combine the conditions
Now we combine the conditions from both functions:
- From \( \csc^{-1}(x) \): \( x \leq -1 \) or \( x \geq 1 \)
- From \( \sin^{-1}\left(\frac{1}{x}\right) \): \( x \geq 1 \)

Thus, the only valid solution for \( x \) that satisfies both conditions is:
\[ x \geq 1 \]

### Conclusion
The values of \( x \) that satisfy the equation \( \csc^{-1}(x) = \sin^{-1}\left(\frac{1}{x}\right) \) are:
\[ x \in [1, \infty) \]