The domain of the function `f(x)=cos^(-1)(sec(cos^-1 x))+sin^(-1)(cosec(sin^(-1)x))` is

Options

A. `x in R`
B. x=1,-1
C. `-1 le x le 1`
D. `x in phi`

Answer

To find the domain of the function \( f(x) = \cos^{-1}(\sec(\cos^{-1} x)) + \sin^{-1}(\csc(\sin^{-1} x)) \), we will analyze each component of the function step by step.

### Step 1: Identify the components of the function
The function consists of two parts:
1. \( \cos^{-1}(\sec(\cos^{-1} x)) \)
2. \( \sin^{-1}(\csc(\sin^{-1} x)) \)

### Step 2: Analyze the first component \( \cos^{-1}(\sec(\cos^{-1} x)) \)
- The domain of \( \cos^{-1}(x) \) is \( x \in [-1, 1] \).
- For \( \sec(\cos^{-1} x) \), we know that \( \sec(\theta) = \frac{1}{\cos(\theta)} \). Therefore, \( \sec(\cos^{-1} x) = \frac{1}{x} \).
- We need \( \frac{1}{x} \) to be in the domain of \( \cos^{-1} \), which means \( \frac{1}{x} \) must lie within the interval \([-1, 1]\).

### Step 3: Set up inequalities for \( \frac{1}{x} \)
We need to solve the inequalities:
1. \( -1 \leq \frac{1}{x} \leq 1 \)

From \( \frac{1}{x} \geq -1 \):
- This implies \( x \leq -1 \) (since \( x \) cannot be zero).

From \( \frac{1}{x} \leq 1 \):
- This implies \( x \geq 1 \).

### Step 4: Analyze the second component \( \sin^{-1}(\csc(\sin^{-1} x)) \)
- The domain of \( \sin^{-1}(x) \) is \( x \in [-1, 1] \).
- For \( \csc(\sin^{-1} x) \), we know that \( \csc(\theta) = \frac{1}{\sin(\theta)} \). Therefore, \( \csc(\sin^{-1} x) = \frac{1}{x} \).
- Similar to the first component, we need \( \frac{1}{x} \) to be in the domain of \( \sin^{-1} \), which means \( \frac{1}{x} \) must also lie within the interval \([-1, 1]\).

### Step 5: Set up inequalities for \( \frac{1}{x} \) in the second component
The inequalities are the same:
1. \( -1 \leq \frac{1}{x} \leq 1 \)

From \( \frac{1}{x} \geq -1 \):
- This implies \( x \leq -1 \).

From \( \frac{1}{x} \leq 1 \):
- This implies \( x \geq 1 \).

### Step 6: Combine the results
From both components, we have:
1. \( x \leq -1 \)
2. \( x \geq 1 \)

### Step 7: Determine the domain
The only values that satisfy both conditions are \( x = -1 \) and \( x = 1 \). Therefore, the domain of the function \( f(x) \) is:
\[ \{ -1, 1 \} \]

### Final Answer
The domain of the function \( f(x) \) is \( x = -1 \) and \( x = 1 \).

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