If `f(x)=sin^(-1)(cosec(sin^(-1)x))+cos^(-1)(sec(cos^(-1)x))`, then f(x) takes

Options

A. exactly two values
B. one value
C. undefined
D. infinite values

Explanation

To solve the problem \( f(x) = \sin^{-1}(\csc(\sin^{-1}(x))) + \cos^{-1}(\sec(\cos^{-1}(x))) \), we will simplify each term step by step.

### Step 1: Simplifying \( \sin^{-1}(\csc(\sin^{-1}(x))) \)

Let \( y = \sin^{-1}(x) \). Then, by definition, we have:
\[ \sin(y) = x \]
The cosecant function is the reciprocal of the sine function:
\[ \csc(y) = \frac{1}{\sin(y)} = \frac{1}{x} \]
Now, we need to find \( \sin^{-1}(\csc(y)) \):
\[ \sin^{-1}(\csc(y)) = \sin^{-1}\left(\frac{1}{x}\right) \]

### Step 2: Simplifying \( \cos^{-1}(\sec(\cos^{-1}(x))) \)

Let \( z = \cos^{-1}(x) \). Then, by definition, we have:
\[ \cos(z) = x \]
The secant function is the reciprocal of the cosine function:
\[ \sec(z) = \frac{1}{\cos(z)} = \frac{1}{x} \]
Now, we need to find \( \cos^{-1}(\sec(z)) \):
\[ \cos^{-1}(\sec(z)) = \cos^{-1}\left(\frac{1}{x}\right) \]

### Step 3: Combining the Results

Now we can combine the results from Step 1 and Step 2:
\[ f(x) = \sin^{-1}\left(\frac{1}{x}\right) + \cos^{-1}\left(\frac{1}{x}\right) \]

### Step 4: Using the Identity

We know that:
\[ \sin^{-1}(a) + \cos^{-1}(a) = \frac{\pi}{2} \quad \text{for } |a| \leq 1 \]
Here, \( a = \frac{1}{x} \). Therefore, we need to ensure that \( |x| \geq 1 \) for \( \frac{1}{x} \) to be valid in the domain of the inverse sine and cosine functions.

### Conclusion

Thus, we conclude:
\[ f(x) = \frac{\pi}{2} \quad \text{if } |x| \geq 1 \]