Find $\frac{d y}{d x}$, if $x=a(\theta+\sin \theta), y=a(1-\cos \theta)$.

Step 1: Differentiate x with respect to θ

We are given $x$ as a function of $\theta$. To find $\frac{d x}{d \theta}$, we differentiate the expression for $x$ with respect to $\theta$. The constant $a$ can be factored out. The derivative of $\theta$ with respect to $\theta$ is 1, and the derivative of $\sin \theta$ with respect to $\theta$ is $\cos \theta$.

Step 2: Calculate dx/dθ

Applying the differentiation rules, we get the derivative of $x$ with respect to $\theta$ as $a(1+\cos \theta)$.

Step 3: Differentiate y with respect to θ

Similarly, we differentiate the expression for $y$ with respect to $\theta$. The constant $a$ is factored out. The derivative of 1 is 0, and the derivative of $-\cos \theta$ is $- (-\sin \theta) = \sin \theta$.

Step 4: Calculate dy/dθ

Applying the differentiation rules, we get the derivative of $y$ with respect to $\theta$ as $a\sin \theta$.

Step 5: Apply the Chain Rule

To find $\frac{d y}{d x}$, we use the chain rule for parametric equations, which states that $\frac{d y}{d x} = \frac{\frac{d y}{d \theta}}{\frac{d x}{d \theta}}$. We substitute the expressions we found for $\frac{d y}{d \theta}$ and $\frac{d x}{d \theta}$.

Step 6: Substitute and Simplify

Substitute the calculated derivatives into the chain rule formula. The constant $a$ cancels out. We can further simplify this expression using trigonometric identities: $\sin \theta = 2\sin(\frac{\theta}{2})\cos(\frac{\theta}{2})$ and $1+\cos \theta = 2\cos^2(\frac{\theta}{2})$.

Step 7: Apply Trigonometric Identities

Using the half-angle identities, $\sin \theta = 2\sin(\frac{\theta}{2})\cos(\frac{\theta}{2})$ and $1+\cos \theta = 2\cos^2(\frac{\theta}{2})$, we substitute these into the expression. The $2$ and one $\cos(\frac{\theta}{2})$ term cancel out, leaving $\frac{\sin(\frac{\theta}{2})}{\cos(\frac{\theta}{2})}$, which simplifies to $\tan(\frac{\theta}{2})$.

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