`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 Inverse Cosecant
The function \( \csc^{-1}(y) \) is defined for \( y \leq -1 \) or \( y \geq 1 \). Since \( x \in \mathbb{R} \setminus (-1, 1) \), it implies that \( -x \) will be either \( \leq -1 \) or \( \geq 1 \) depending on the value of \( x \).

### Step 2: Rewrite the Expression
Using the property of inverse trigonometric functions, we know that:
\[ \csc^{-1}(-x) = -\csc^{-1}(x) \]
This is because the cosecant function is an odd function, meaning that \( \csc(-\theta) = -\csc(\theta) \).

### Step 3: Convert to Inverse Sine
Recall that:
\[ \csc^{-1}(x) = \sin^{-1}\left(\frac{1}{x}\right) \]
Thus, we can express \( -\csc^{-1}(x) \) as:
\[ -\csc^{-1}(x) = -\sin^{-1}\left(\frac{1}{x}\right) \]

### Step 4: Combine the Results
Putting it all together, we have:
\[ \csc^{-1}(-x) = -\sin^{-1}\left(\frac{1}{x}\right) \]

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

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