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

Explanation

To find \(\frac{dy}{dx}\) given the equations \(x = \cos \theta - \cos 2\theta\) and \(y = \sin \theta - \sin 2\theta\), we will use the chain rule of differentiation. Here’s the step-by-step solution:

### Step 1: Differentiate \(x\) with respect to \(\theta\)
Given:
\[ x = \cos \theta - \cos 2\theta \]
Using the chain rule and the derivative of cosine:
\[ \frac{dx}{d\theta} = -\sin \theta - (-\sin 2\theta \cdot 2) = -\sin \theta + 2\sin 2\theta \]
Thus,
\[ \frac{dx}{d\theta} = 2\sin 2\theta - \sin \theta \]

### Step 2: Differentiate \(y\) with respect to \(\theta\)
Given:
\[ y = \sin \theta - \sin 2\theta \]
Using the chain rule and the derivative of sine:
\[ \frac{dy}{d\theta} = \cos \theta - (2\cos 2\theta) \]
Thus,
\[ \frac{dy}{d\theta} = \cos \theta - 2\cos 2\theta \]

### Step 3: Find \(\frac{dy}{dx}\)
Using the chain rule:
\[ \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = \frac{\cos \theta - 2\cos 2\theta}{2\sin 2\theta - \sin \theta} \]

### Final Answer
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \frac{\cos \theta - 2\cos 2\theta}{2\sin 2\theta - \sin \theta} \]

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