If `y=cot^(-1)sqrt((1-sinx)/(1+sinx)),` find `(dy)/(dx).`

Options

A. `-(1)/(3)`
B. `-(1)/(2)`
C. `(1)/(3)`
D. `(1)/(2)`

Explanation

To solve the problem, we need to differentiate the function \( y = \cot^{-1} \left( \sqrt{\frac{1 - \sin x}{1 + \sin x}} \right) \).

### Step 1: Simplify the Argument
First, we can simplify the expression inside the cotangent inverse. We have:

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

Using the identity \( \frac{1 - \sin x}{1 + \sin x} = \frac{\cos^2 \frac{x}{2}}{\sin^2 \frac{x}{2}} = \cot^2 \frac{x}{2} \), we can rewrite:

\[ y = \cot^{-1} \left( \cot \frac{x}{2} \right) \]

### Step 2: Use the Inverse Cotangent Identity
From the properties of inverse functions, we know that:

\[ \cot^{-1}(\cot \theta) = \theta \quad \text{(for } \theta \text{ in the appropriate range)} \]

Thus, we can simplify \( y \) to:

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

### Step 3: Differentiate with Respect to \( x \)
Now, we differentiate \( y \) with respect to \( x \):

\[ \frac{dy}{dx} = \frac{1}{2} \]

### Final Answer
Thus, the derivative of \( y \) with respect to \( x \) is:

\[ \frac{dy}{dx} = \frac{1}{2} \]

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