Find `(dy)/(dx)` if x= 3 cos theta - cos 2theta and y= sin theta - sin 2theta.`

Explanation

To find \(\frac{dy}{dx}\) given \(x = 3 \cos \theta - \cos 2\theta\) and \(y = \sin \theta - \sin 2\theta\), we will follow these steps:

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

Given:
\[ x = 3 \cos \theta - \cos 2\theta \]

Using the chain rule, we differentiate \(x\):
\[ \frac{dx}{d\theta} = \frac{d}{d\theta}(3 \cos \theta) - \frac{d}{d\theta}(\cos 2\theta) \]

Calculating each derivative:
1. The derivative of \(3 \cos \theta\) is \(-3 \sin \theta\).
2. The derivative of \(\cos 2\theta\) using the chain rule is \(-\sin 2\theta \cdot 2 = -2 \sin 2\theta\).

Combining these results:
\[ \frac{dx}{d\theta} = -3 \sin \theta + 2 \sin 2\theta \]

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

Given:
\[ y = \sin \theta - \sin 2\theta \]

Using the chain rule, we differentiate \(y\):
\[ \frac{dy}{d\theta} = \frac{d}{d\theta}(\sin \theta) - \frac{d}{d\theta}(\sin 2\theta) \]

Calculating each derivative:
1. The derivative of \(\sin \theta\) is \(\cos \theta\).
2. The derivative of \(\sin 2\theta\) using the chain rule is \(\cos 2\theta \cdot 2 = 2 \cos 2\theta\).

Combining these results:
\[ \frac{dy}{d\theta} = \cos \theta - 2 \cos 2\theta \]

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

To find \(\frac{dy}{dx}\), we use the formula:
\[ \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} \]

Substituting the derivatives we found:
\[ \frac{dy}{dx} = \frac{\cos \theta - 2 \cos 2\theta}{-3 \sin \theta + 2 \sin 2\theta} \]

### Final Result

Thus, the final expression for \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \frac{\cos \theta - 2 \cos 2\theta}{-3 \sin \theta + 2 \sin 2\theta} \]

---