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

Options

A. `"cot"(theta)/2`
B. `"tan"(theta)/2`
C. `1/2"cosec^(2)(theta)/2`
D. `-1/2 "cosec^(2)(theta)/2`

Explanation

To find \(\frac{dy}{dx}\) given \(x = -a(\theta - \sin \theta)\) and \(y = a(1 - \cos \theta)\), we can use the chain rule for derivatives in parametric equations.

### Step 1: Find \(\frac{dx}{d\theta}\)

Given:
\[ x = -a(\theta - \sin \theta) \]

To differentiate \(x\) with respect to \(\theta\):
\[ \frac{dx}{d\theta} = -a\left(1 - \cos \theta\right) \]

### Step 2: Find \(\frac{dy}{d\theta}\)

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

To differentiate \(y\) with respect to \(\theta\):
\[ \frac{dy}{d\theta} = a\sin \theta \]

### Step 3: Find \(\frac{dy}{dx}\)

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

Substituting the values from Steps 1 and 2:
\[ \frac{dy}{dx} = \frac{a\sin \theta}{-a(1 - \cos \theta)} \]

### Step 4: Simplify the expression

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

### Step 5: Further simplification using trigonometric identities

We can use the identity \(1 - \cos \theta = 2\sin^2(\frac{\theta}{2})\) and \(\sin \theta = 2\sin(\frac{\theta}{2})\cos(\frac{\theta}{2})\):
\[ \frac{dy}{dx} = -\frac{2\sin(\frac{\theta}{2})\cos(\frac{\theta}{2})}{2\sin^2(\frac{\theta}{2})} = -\frac{\cos(\frac{\theta}{2})}{\sin(\frac{\theta}{2})} = -\cot\left(\frac{\theta}{2}\right) \]

### Final Result:
\[ \frac{dy}{dx} = -\cot\left(\frac{\theta}{2}\right) \]