The derivative of `cos(2tan^(-1)sqrt((1-x)/(1+x)))-2cos^(-1)sqrt((1-x)/(2)))` w.r.t. x is

Options

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

Explanation

To find the derivative of the expression \( y = \cos(2 \tan^{-1}(\sqrt{\frac{1-x}{1+x}})) - 2 \cos^{-1}(\sqrt{\frac{1-x}{2}}) \) with respect to \( x \), we will follow these steps:

### Step 1: Substitute \( x \)
Let \( x = \cos(\theta) \). Then we can express \( \frac{1-x}{1+x} \) in terms of \( \theta \):
\[ \frac{1 - \cos(\theta)}{1 + \cos(\theta)} = \frac{2 \sin^2(\theta/2)}{2 \cos^2(\theta/2)} = \tan^2(\theta/2) \]
Thus, we have:
\[ \tan^{-1}(\sqrt{\frac{1-x}{1+x}}) = \tan^{-1}(\tan(\theta/2)) = \frac{\theta}{2} \]

### Step 2: Simplify the expression for \( y \)
Now substituting back into \( y \):
\[ y = \cos(2 \cdot \frac{\theta}{2}) - 2 \cos^{-1}(\sqrt{\frac{1 - \cos(\theta)}{2}}) \]
Using the double angle identity for cosine:
\[ y = \cos(\theta) - 2 \cos^{-1}(\sin(\theta/2)) \]

### Step 3: Rewrite \( \cos^{-1} \)
Using the identity \( \cos^{-1}(x) = \frac{\pi}{2} - \sin^{-1}(x) \):
\[ y = \cos(\theta) - 2 \left( \frac{\pi}{2} - \sin^{-1}(\sin(\theta/2)) \right) \]
This simplifies to:
\[ y = \cos(\theta) - \pi + 2 \cdot \frac{\theta}{2} = \cos(\theta) - \pi + \theta \]

### Step 4: Differentiate \( y \)
Now, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(\cos(\theta) - \pi + \theta) \]
Using the chain rule:
\[ \frac{dy}{dx} = -\sin(\theta) \cdot \frac{d\theta}{dx} + \frac{d\theta}{dx} \]
We know that:
\[ \frac{d\theta}{dx} = -\frac{1}{\sqrt{1 - x^2}} \]
Thus:
\[ \frac{dy}{dx} = -\sin(\theta) \cdot \left(-\frac{1}{\sqrt{1 - x^2}}\right) + \left(-\frac{1}{\sqrt{1 - x^2}}\right) \]
This simplifies to:
\[ \frac{dy}{dx} = \frac{\sin(\theta) - 1}{\sqrt{1 - x^2}} \]

### Step 5: Substitute back \( \sin(\theta) \)
Since \( x = \cos(\theta) \), we can find \( \sin(\theta) \) using:
\[ \sin(\theta) = \sqrt{1 - x^2} \]
Thus:
\[ \frac{dy}{dx} = \frac{\sqrt{1 - x^2} - 1}{\sqrt{1 - x^2}} \]

### Final Answer
The derivative of the given expression is:
\[ \frac{dy}{dx} = 1 - \frac{1}{\sqrt{1 - x^2}} \]