If `f (x) = sqrt((1+ sin ^(-1) x)/(1- tan ^(-1)x)),` then f (0) is equal to :

Options

A. 4
B. 3
C. 2
D. 1

Answer

To find \( f(0) \) for the function \( f(x) = \sqrt{\frac{1 + \sin^{-1} x}{1 - \tan^{-1} x}} \), we will follow these steps:

### Step 1: Substitute \( x = 0 \) into the function

We start by substituting \( x = 0 \) into the function:

\[ f(0) = \sqrt{\frac{1 + \sin^{-1}(0)}{1 - \tan^{-1}(0)}} \]

### Step 2: Evaluate \( \sin^{-1}(0) \)

The value of \( \sin^{-1}(0) \) is:

\[ \sin^{-1}(0) = 0 \]

### Step 3: Evaluate \( \tan^{-1}(0) \)

The value of \( \tan^{-1}(0) \) is:

\[ \tan^{-1}(0) = 0 \]

### Step 4: Substitute these values back into the function

Now we substitute these values back into the function:

\[ f(0) = \sqrt{\frac{1 + 0}{1 - 0}} = \sqrt{\frac{1}{1}} = \sqrt{1} \]

### Step 5: Simplify the expression

Finally, we simplify the expression:

\[ f(0) = 1 \]

Thus, the value of \( f(0) \) is:

\[ \boxed{1} \]

---