If `x = 2(theta + sin theta) and y = 2(1 - cos theta)`, then value of `(dy)/(dx)` is

Options

A. `tan(theta/2)`
B. `cot (theta/2)`
C. `sin (theta/2)`
D. `cos (theta/2)`

Answer

To find the value of \(\frac{dy}{dx}\) given the equations \(x = 2(\theta + \sin \theta)\) and \(y = 2(1 - \cos \theta)\), we will use the chain rule of differentiation. Here’s a step-by-step solution:

### Step 1: Differentiate \(y\) with respect to \(\theta\)

Given:
\[ y = 2(1 - \cos \theta) \]

Differentiating \(y\) with respect to \(\theta\):
\[ \frac{dy}{d\theta} = 2 \cdot \frac{d}{d\theta}(1 - \cos \theta) = 2 \cdot \sin \theta \]

### Step 2: Differentiate \(x\) with respect to \(\theta\)

Given:
\[ x = 2(\theta + \sin \theta) \]

Differentiating \(x\) with respect to \(\theta\):
\[ \frac{dx}{d\theta} = 2 \cdot \frac{d}{d\theta}(\theta + \sin \theta) = 2(1 + \cos \theta) \]

### Step 3: Use the chain rule to find \(\frac{dy}{dx}\)

Using the chain rule:
\[ \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} \]

Substituting the derivatives we found:
\[ \frac{dy}{dx} = \frac{2 \sin \theta}{2(1 + \cos \theta)} = \frac{\sin \theta}{1 + \cos \theta} \]

### Step 4: Simplify the expression

We can simplify \(\frac{\sin \theta}{1 + \cos \theta}\) using the double angle identity:
\[ \frac{\sin \theta}{1 + \cos \theta} = \tan\left(\frac{\theta}{2}\right) \]

Thus, the final result is:
\[ \frac{dy}{dx} = \tan\left(\frac{\theta}{2}\right) \]

### Final Answer:
\[ \frac{dy}{dx} = \tan\left(\frac{\theta}{2}\right) \]

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