Find $\frac{d y}{d x}$, if $x=a \cos \theta, y=a \sin \theta$.

Step 1: Differentiate x with respect to θ

We are given $x$ as a function of $\theta$. To find $\frac{dx}{d\theta}$, we differentiate $a \cos \theta$ with respect to $\theta$. The derivative of $\cos \theta$ is $-\sin \theta$, so $\frac{dx}{d\theta} = -a \sin \theta$.

Step 2: Differentiate y with respect to θ

Similarly, we differentiate $y$ with respect to $\theta$. The derivative of $a \sin \theta$ with respect to $\theta$ is $a \cos \theta$, since the derivative of $\sin \theta$ is $\cos \theta$.

Step 3: Apply the Chain Rule

To find $\frac{dy}{dx}$, we use the chain rule for parametric equations. This rule states that $\frac{dy}{dx}$ can be found by dividing $\frac{dy}{d\theta}$ by $\frac{dx}{d\theta}$.

Step 4: Substitute and Simplify

Substitute the expressions for $\frac{dy}{d\theta}$ and $\frac{dx}{d\theta}$ that we found in the previous steps. The $a$ terms cancel out, and $\frac{\cos \theta}{\sin \theta}$ simplifies to $\cot \theta$, resulting in $-\cot \theta$.

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