`cosec^(-1)(-x),x in R-(-1,1),` is equal to

Options

A. `cosec^(-1)x`
B. `-sin^(-1)x`
C. `-sin^(-1)(1/x)`
D. `pi-cosec^(-1)x`

Answer

To solve the expression \( \csc^{-1}(-x) \) where \( x \in \mathbb{R} \setminus (-1, 1) \), we can follow these steps:

### Step 1: Understand the definition of the cosecant inverse function
The cosecant inverse function, \( \csc^{-1}(y) \), is defined for \( y \leq -1 \) or \( y \geq 1 \). Since \( x \) is in \( \mathbb{R} \setminus (-1, 1) \), it implies that \( -x \) will be either \( \leq -1 \) or \( \geq 1 \).

### Step 2: Rewrite the cosecant inverse in terms of sine
Using the identity \( \csc^{-1}(y) = \sin^{-1}\left(\frac{1}{y}\right) \), we can rewrite \( \csc^{-1}(-x) \) as:
\[ \csc^{-1}(-x) = \sin^{-1}\left(\frac{1}{-x}\right) = -\sin^{-1}\left(\frac{1}{x}\right) \]

### Step 3: Consider the range of \( x \)
Since \( x \in \mathbb{R} \setminus (-1, 1) \), it means \( x \) can take values less than -1 or greater than 1. This ensures that \( \frac{1}{x} \) will also be outside the interval (-1, 1).

### Step 4: Determine the final expression
Thus, we conclude that:
\[ \csc^{-1}(-x) = -\sin^{-1}\left(\frac{1}{x}\right) \]

### Final Result
The expression \( \csc^{-1}(-x) \) is equal to:
\[ -\sin^{-1}\left(\frac{1}{x}\right) \]

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