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 follow these steps:

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

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

Differentiating \(y\) with respect to \(\theta\):
\[ \frac{dy}{d\theta} = 2 \cdot \frac{d}{d\theta}(1 - \cos \theta) \]
Since the derivative of a constant is 0 and the derivative of \(-\cos \theta\) is \(\sin \theta\):
\[ \frac{dy}{d\theta} = 2 \cdot \sin \theta \]

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

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

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

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

Using the chain rule, we have:
\[ \frac{dy}{dx} = \frac{dy}{d\theta} \cdot \frac{d\theta}{dx} \]
This can be rewritten as:
\[ \frac{dy}{dx} = \frac{dy}{d\theta} \cdot \frac{1}{\frac{dx}{d\theta}} \]

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

### Step 4: Simplify the expression

The \(2\) in the numerator and denominator cancels out:
\[ \frac{dy}{dx} = \frac{\sin \theta}{1 + \cos \theta} \]

### Step 5: Convert to half-angle identities

Using the half-angle identities:
\[ \sin \theta = 2 \sin\left(\frac{\theta}{2}\right) \cos\left(\frac{\theta}{2}\right) \]
\[ 1 + \cos \theta = 2 \cos^2\left(\frac{\theta}{2}\right) \]

Substituting these into our expression:
\[ \frac{dy}{dx} = \frac{2 \sin\left(\frac{\theta}{2}\right) \cos\left(\frac{\theta}{2}\right)}{2 \cos^2\left(\frac{\theta}{2}\right)} \]

The \(2\) cancels out:
\[ \frac{dy}{dx} = \frac{\sin\left(\frac{\theta}{2}\right)}{\cos\left(\frac{\theta}{2}\right)} = \tan\left(\frac{\theta}{2}\right) \]

### Final Result
Thus, the value of \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \tan\left(\frac{\theta}{2}\right) \]