If ` Y= sec^(-1)"" (sqrt""x+1)/( sqrt""x-1 )) + sin^(-1)"" (sqrt""x-1)/( sqrt""x+1) `, then ` (dy)/(dx)=`

Options

A. 1
B. 2
C. 3
D. 0

Answer

To solve the problem, we need to differentiate the given function:

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

### Step 1: Simplify the Expression
We can use the identity for inverse trigonometric functions:
\[ \sin^{-1}(y) + \cos^{-1}(y) = \frac{\pi}{2} \]

Notice that:
- Let \( y = \frac{\sqrt{x}-1}{\sqrt{x}+1} \)
- Then, \( \sec^{-1}\left(\frac{\sqrt{x}+1}{\sqrt{x}-1}\right) \) can be rewritten using the reciprocal relation.

Using the identity:
\[ \sec^{-1}(z) = \cos^{-1}\left(\frac{1}{z}\right) \]

We have:
\[ \sec^{-1}\left(\frac{\sqrt{x}+1}{\sqrt{x}-1}\right) = \cos^{-1}\left(\frac{\sqrt{x}-1}{\sqrt{x}+1}\right) \]

Thus, we can rewrite \( Y \) as:
\[ Y = \cos^{-1}\left(\frac{\sqrt{x}-1}{\sqrt{x}+1}\right) + \sin^{-1}\left(\frac{\sqrt{x}-1}{\sqrt{x}+1}\right) \]

### Step 2: Apply the Identity
Now, applying the identity:
\[ Y = \frac{\pi}{2} \]

### Step 3: Differentiate
Since \( Y \) is a constant (\( \frac{\pi}{2} \)), its derivative with respect to \( x \) is:
\[ \frac{dY}{dx} = 0 \]

### Final Result
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = 0 \]

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