If `y=cos^(-1)((sqrt(x)-1)/(sqrt(x)+1))+cosec^(-1)((sqrt(x)+1)/(sqrt(x)-1))` , then `(dy)/(dx)` is equal to

Options

A. `(pi)/(2)`
B. 0
C. 1
D. none of these

Answer

To find the derivative \( \frac{dy}{dx} \) for the given function

\[ y = \cos^{-1}\left(\frac{\sqrt{x}-1}{\sqrt{x}+1}\right) + \csc^{-1}\left(\frac{\sqrt{x}+1}{\sqrt{x}-1}\right), \]

we can simplify the expression using properties of inverse trigonometric functions.

### Step 1: Recognize the relationship between inverse trigonometric functions
We know that:

\[ \cos^{-1}(a) + \sin^{-1}(a) = \frac{\pi}{2} \]

for any \( a \) in the domain of these functions.

### Step 2: Rewrite the expression
Notice that:

\[ \csc^{-1}(x) = \sin^{-1}\left(\frac{1}{x}\right) \]

Thus, we can rewrite \( \csc^{-1}\left(\frac{\sqrt{x}+1}{\sqrt{x}-1}\right) \) as:

\[ \sin^{-1}\left(\frac{1}{\frac{\sqrt{x}+1}{\sqrt{x}-1}}\right) = \sin^{-1}\left(\frac{\sqrt{x}-1}{\sqrt{x}+1}\right) \]

### Step 3: Combine the expressions
Now we can combine the two parts of \( y \):

\[ y = \cos^{-1}\left(\frac{\sqrt{x}-1}{\sqrt{x}+1}\right) + \sin^{-1}\left(\frac{\sqrt{x}-1}{\sqrt{x}+1}\right) \]

Using the identity from Step 1, we have:

\[ y = \frac{\pi}{2} \]

### Step 4: Differentiate
Since \( y \) is a constant (\( \frac{\pi}{2} \)), we find the derivative:

\[ \frac{dy}{dx} = 0 \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is equal to:

\[ \boxed{0} \]