Find `(dy)/(dx)` : `x=a{cost+1/2logt a n^2t/2}` and `y=asint` .

Explanation

To find \(\frac{dy}{dx}\) for the given equations \(x = a \cos t + \frac{1}{2} \log \tan^2 \frac{t}{2}\) and \(y = a \sin t\), we can use the chain rule. We will first find \(\frac{dy}{dt}\) and \(\frac{dx}{dt}\), and then use these to find \(\frac{dy}{dx}\).

### Step 1: Differentiate \(y\) with respect to \(t\)
Given:
\[ y = a \sin t \]

Differentiating with respect to \(t\):
\[
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