If `y="cosec"^(-1)((x^(2)+1)/(x^(2)-1))+cos^(-1) ((x^(2)-1)/(x^(2)+1)),"then " (dy)/(dx)=`

Options

A. ` 0`
B. ` 1`
C. ` (1)/(sqrt( 1-x^(2)))`
D. ` (-1)/(sqrt( 1-x^(2)))`

Answer

To solve the problem, we need to differentiate the given function \( y = \csc^{-1} \left( \frac{x^2 + 1}{x^2 - 1} \right) + \cos^{-1} \left( \frac{x^2 - 1}{x^2 + 1} \right) \).

### Step 1: Rewrite the Inverse Functions
Using the identity for inverse trigonometric functions, we can rewrite the equation:
\[ y = \csc^{-1} \left( \frac{x^2 + 1}{x^2 - 1} \right) + \cos^{-1} \left( \frac{x^2 - 1}{x^2 + 1} \right) \]
We know that:
\[ \csc^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2} \]
Thus, we can express \( y \) as:
\[ y = \frac{\pi}{2} \]

### Step 2: Differentiate the Function
Now we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx} \left( \frac{\pi}{2} \right) \]
Since \( \frac{\pi}{2} \) is a constant, its derivative is:
\[ \frac{dy}{dx} = 0 \]

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