`x=a (t-sin t) , y =a (1-cos t) `

Answer

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

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

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

We differentiate \(x\) with respect to \(t\):
\[ \frac{dx}{dt} = a\left(\frac{d}{dt}(t) - \frac{d}{dt}(\sin t)\right) = a(1 - \cos t) \]

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

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

We differentiate \(y\) with respect to \(t\):
\[ \frac{dy}{dt} = a\left(0 - \frac{d}{dt}(\cos t)\right) = a\sin t \]

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

Using the chain rule, we can find \(\frac{dy}{dx}\) as follows:
\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \]

Substituting the derivatives we found:
\[ \frac{dy}{dx} = \frac{a\sin t}{a(1 - \cos t)} = \frac{\sin t}{1 - \cos t} \]

### Step 4: Simplify the expression

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

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

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