If `y=tan^(-1) ((sin x)/(1-cos x)),"then " dy/dx=`

Options

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

Answer

To find the derivative of the function \( y = \tan^{-1} \left( \frac{\sin x}{1 - \cos x} \right) \), we will use the chain rule and the quotient rule. Let's go through the steps:

### Step 1: Differentiate \( y \)
We start by applying the derivative of the inverse tangent function:
\[ \frac{dy}{dx} = \frac{1}{1 + \left( \frac{\sin x}{1 - \cos x} \right)^2} \cdot \frac{d}{dx} \left( \frac{\sin x}{1 - \cos x} \right) \]

### Step 2: Differentiate \( \frac{\sin x}{1 - \cos x} \)
To differentiate \( \frac{\sin x}{1 - \cos x} \), we will use the quotient rule:
\[ \frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{g(x) f'(x) - f(x) g'(x)}{(g(x))^2} \]
where \( f(x) = \sin x \) and \( g(x) = 1 - \cos x \).

Calculating the derivatives:
- \( f'(x) = \cos x \)
- \( g'(x) = \sin x \)

Now applying the quotient rule:
\[ \frac{d}{dx} \left( \frac{\sin x}{1 - \cos x} \right) = \frac{(1 - \cos x) \cos x - \sin x \cdot \sin x}{(1 - \cos x)^2} \]
This simplifies to:
\[ = \frac{(1 - \cos x) \cos x - \sin^2 x}{(1 - \cos x)^2} \]

### Step 3: Simplify the numerator
Using the identity \( \sin^2 x + \cos^2 x = 1 \):
\[ = \frac{\cos x - \cos^2 x - \sin^2 x}{(1 - \cos x)^2} = \frac{\cos x - 1}{(1 - \cos x)^2} \]

### Step 4: Substitute back into the derivative
Now we substitute this result back into our expression for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{1}{1 + \left( \frac{\sin x}{1 - \cos x} \right)^2} \cdot \frac{\cos x - 1}{(1 - \cos x)^2} \]

### Step 5: Simplify \( 1 + \left( \frac{\sin x}{1 - \cos x} \right)^2 \)
Calculating \( 1 + \left( \frac{\sin x}{1 - \cos x} \right)^2 \):
\[ = 1 + \frac{\sin^2 x}{(1 - \cos x)^2} = \frac{(1 - \cos x)^2 + \sin^2 x}{(1 - \cos x)^2} \]
Using \( \sin^2 x + \cos^2 x = 1 \):
\[ = \frac{1 - 2\cos x + \cos^2 x + \sin^2 x}{(1 - \cos x)^2} = \frac{1 - 2\cos x + 1}{(1 - \cos x)^2} = \frac{2 - 2\cos x}{(1 - \cos x)^2} = \frac{2(1 - \cos x)}{(1 - \cos x)^2} = \frac{2}{1 - \cos x} \]

### Step 6: Final expression for \( \frac{dy}{dx} \)
Substituting this back into our expression:
\[ \frac{dy}{dx} = \frac{(1 - \cos x)(\cos x - 1)}{(1 - \cos x)^2 \cdot \frac{2}{1 - \cos x}} = \frac{(\cos x - 1)}{2(1 - \cos x)} = -\frac{1}{2} \]

### Final Answer:
\[ \frac{dy}{dx} = -\frac{1}{2} \]