If `x=a ( theta -sin theta ) ,y=a(1-cos theta ) ,then (dy)/(dx) =`

Options

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

Answer

To find \(\frac{dy}{dx}\) given the parametric equations \(x = a(\theta - \sin \theta)\) and \(y = a(1 - \cos \theta)\), we will follow these steps:

### Step 1: Differentiate \(x\) with respect to \(\theta\)
We start by differentiating \(x\) with respect to \(\theta\):
\[ \frac{dx}{d\theta} = \frac{d}{d\theta}[a(\theta - \sin \theta)] = a\left(1 - \cos \theta\right) \]
**Hint:** Remember that the derivative of \(\sin \theta\) is \(\cos \theta\).

### Step 2: Differentiate \(y\) with respect to \(\theta\)
Next, we differentiate \(y\) with respect to \(\theta\):
\[ \frac{dy}{d\theta} = \frac{d}{d\theta}[a(1 - \cos \theta)] = a\left(0 + \sin \theta\right) = a \sin \theta \]
**Hint:** The derivative of \(-\cos \theta\) is \(\sin \theta\).

### Step 3: Find \(\frac{dy}{dx}\)
Now, we can find \(\frac{dy}{dx}\) using the chain rule:
\[ \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} = \frac{a \sin \theta}{a(1 - \cos \theta)} \]
**Hint:** When dividing, the \(a\) terms will cancel out.

### Step 4: Simplify the expression
After canceling \(a\):
\[ \frac{dy}{dx} = \frac{\sin \theta}{1 - \cos \theta} \]
**Hint:** You can use trigonometric identities to simplify further.

### Step 5: Use trigonometric identities
Using the identity \(1 - \cos \theta = 2 \sin^2(\theta/2)\) and \(\sin \theta = 2 \sin(\theta/2) \cos(\theta/2)\):
\[ \frac{dy}{dx} = \frac{2 \sin(\theta/2) \cos(\theta/2)}{2 \sin^2(\theta/2)} = \frac{\cos(\theta/2)}{\sin(\theta/2)} = \cot(\theta/2) \]
**Hint:** Remember that \(\cot\) is the ratio of \(\cos\) to \(\sin\).

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