`(d)/(dx)[ sin^2 cot^(-1) "" sqrt((1-x)/(1+x))]` equals

Options

A. `-1/2`
B. `1/2`
C. `1`
D. `-1`

Answer

To solve the problem \(\frac{d}{dx}\left[ \sin^2\left(\cot^{-1}\left(\sqrt{\frac{1-x}{1+x}}\right)\right)\right]\), we will use the chain rule and some trigonometric identities. Let's go through the solution step by step.

### Step 1: Rewrite the Function
We start with the function:
\[ y = \sin^2\left(\cot^{-1}\left(\sqrt{\frac{1-x}{1+x}}\right)\right) \]

### Step 2: Use the Identity for Sine
Using the identity \(\sin^2(\theta) = \frac{1}{\csc^2(\theta)}\), we can rewrite the function as:
\[ y = \frac{1}{\csc^2\left(\cot^{-1}\left(\sqrt{\frac{1-x}{1+x}}\right)\right)} \]

### Step 3: Apply the Cosecant Identity
Recall that \(\csc^2(\theta) = 1 + \cot^2(\theta)\). Therefore, we have:
\[ y = \frac{1}{1 + \cot^2\left(\cot^{-1}\left(\sqrt{\frac{1-x}{1+x}}\right)\right)} \]

### Step 4: Simplify Cotangent
Since \(\cot(\cot^{-1}(x)) = x\), we can simplify:
\[ y = \frac{1}{1 + \left(\sqrt{\frac{1-x}{1+x}}\right)^2} \]

### Step 5: Simplify the Expression
Now, we simplify \(\left(\sqrt{\frac{1-x}{1+x}}\right)^2\):
\[ y = \frac{1}{1 + \frac{1-x}{1+x}} = \frac{1}{\frac{(1+x) + (1-x)}{1+x}} = \frac{1+x}{2} \]

### Step 6: Differentiate
Now we differentiate \(y\) with respect to \(x\):
\[ \frac{dy}{dx} = \frac{d}{dx}\left(\frac{1+x}{2}\right) = \frac{1}{2} \]

### Final Answer
Thus, the derivative is:
\[ \frac{d}{dx}\left[ \sin^2\left(\cot^{-1}\left(\sqrt{\frac{1-x}{1+x}}\right)\right)\right] = \frac{1}{2} \]