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

Options

A. `"tan"(t)/(2)`
B. ` cot (t)/(2)`
C. `sec (t)/(2)`
D. `"cosec"(t)/(2)`

Answer

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

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

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

Differentiating \( y \) with respect to \( t \):
\[ \frac{dy}{dt} = a \cdot \frac{d}{dt}(1 - \cos t) = a \cdot (0 + \sin t) = a \sin t \]

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

Given:
\[ x = a(t + \sin t) \]

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

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

Using the chain rule for parametric equations:
\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \]

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

### Step 4: Simplify the expression

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

### Step 5: Further simplification using trigonometric identities

Using the identity \( 1 + \cos t = 2 \cos^2\left(\frac{t}{2}\right) \) and \( \sin t = 2 \sin\left(\frac{t}{2}\right) \cos\left(\frac{t}{2}\right) \):
\[ \frac{dy}{dx} = \frac{2 \sin\left(\frac{t}{2}\right) \cos\left(\frac{t}{2}\right)}{2 \cos^2\left(\frac{t}{2}\right)} = \frac{\sin\left(\frac{t}{2}\right)}{\cos\left(\frac{t}{2}\right)} = \tan\left(\frac{t}{2}\right) \]

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

---