If `x=a(t-sint), y=a(1-cost)` then find `(d^2y)/(dx^2)`.

Answer

To find \(\frac{d^2y}{dx^2}\) given the parametric equations \(x = a(t - \sin t)\) and \(y = a(1 - \cos t)\), we will follow these steps:

### Step 1: Find \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\)

1. Differentiate \(x\) with respect to \(t\):
\[ \frac{dx}{dt} = a\left(1 - \cos t\right) \]
Here, the derivative of \(t\) is \(1\) and the derivative of \(-\sin t\) is \(-\cos t\).

2. Differentiate \(y\) with respect to \(t\):
\[ \frac{dy}{dt} = a\sin t \]
The derivative of \(1\) is \(0\) and the derivative of \(-\cos t\) is \(\sin t\).

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

Using the chain rule:
\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{a\sin t}{a(1 - \cos t)} \]
The \(a\) cancels out:
\[ \frac{dy}{dx} = \frac{\sin t}{1 - \cos t} \]

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

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

### Step 4: Find \(\frac{d^2y}{dx^2}\)

Now, we differentiate \(\frac{dy}{dx}\) with respect to \(x\):
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\cot\left(\frac{t}{2}\right)\right) \]
Using the chain rule:
\[ \frac{d}{dx}\left(\cot\left(\frac{t}{2}\right)\right) = \frac{d}{dt}\left(\cot\left(\frac{t}{2}\right)\right) \cdot \frac{dt}{dx} \]

### Step 5: Find \(\frac{d}{dt}\left(\cot\left(\frac{t}{2}\right)\right)\)

Using the derivative of \(\cot x\):
\[ \frac{d}{dt}\left(\cot\left(\frac{t}{2}\right)\right) = -\csc^2\left(\frac{t}{2}\right) \cdot \frac{1}{2} = -\frac{1}{2}\csc^2\left(\frac{t}{2}\right) \]

### Step 6: Find \(\frac{dt}{dx}\)

From \(\frac{dx}{dt} = a(1 - \cos t)\):
\[ \frac{dt}{dx} = \frac{1}{\frac{dx}{dt}} = \frac{1}{a(1 - \cos t)} \]

### Step 7: Combine the results

Now we can substitute back:
\[ \frac{d^2y}{dx^2} = -\frac{1}{2}\csc^2\left(\frac{t}{2}\right) \cdot \frac{1}{a(1 - \cos t)} \]

### Final Result

Thus, we have:
\[ \frac{d^2y}{dx^2} = -\frac{\csc^2\left(\frac{t}{2}\right)}{2a(1 - \cos t)} \]