If `x=2 cos theta- cos 2 theta and y=2 sin theta - sin 2 theta,` then `(dy)/(dx)`=

Options

A. `- tan ((3theta)/2)`
B. `cot(theta/2)`
C. `tan ((3theta)/2)`
D. `cot ((2theta)/3)`

Answer

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

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

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

Differentiate \(x\) with respect to \(\theta\):
\[ \frac{dx}{d\theta} = \frac{d}{d\theta}(2 \cos \theta) - \frac{d}{d\theta}(\cos 2\theta) \]
Using the derivative of \(\cos\) and the chain rule:
\[ \frac{dx}{d\theta} = -2 \sin \theta + 2 \sin 2\theta \]

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

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

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

### Step 3: Use the chain rule to find \(\frac{dy}{dx}\)

By the chain rule:
\[ \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} \]

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

### Step 4: Simplify the expression

Factor out common terms:
\[ \frac{dy}{dx} = \frac{2(\cos \theta - \cos 2\theta)}{2(-\sin \theta + \sin 2\theta)} \]
This simplifies to:
\[ \frac{dy}{dx} = \frac{\cos \theta - \cos 2\theta}{-\sin \theta + \sin 2\theta} \]

### Step 5: Use trigonometric identities

Using the identities:
\[ \cos A - \cos B = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \]
and
\[ \sin A - \sin B = 2 \cos\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \]

Let \(A = 2\theta\) and \(B = \theta\):
\[ \cos \theta - \cos 2\theta = -2 \sin\left(\frac{3\theta}{2}\right) \sin\left(\frac{\theta}{2}\right) \]
\[ -\sin \theta + \sin 2\theta = 2 \cos\left(\frac{3\theta}{2}\right) \sin\left(\frac{\theta}{2}\right) \]

Substituting these into our expression for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{-2 \sin\left(\frac{3\theta}{2}\right) \sin\left(\frac{\theta}{2}\right)}{2 \cos\left(\frac{3\theta}{2}\right) \sin\left(\frac{\theta}{2}\right)} \]

### Step 6: Cancel common terms

Cancel \(2\) and \(\sin\left(\frac{\theta}{2}\right)\) (assuming \(\sin\left(\frac{\theta}{2}\right) \neq 0\)):
\[ \frac{dy}{dx} = -\frac{\sin\left(\frac{3\theta}{2}\right)}{\cos\left(\frac{3\theta}{2}\right)} = -\tan\left(\frac{3\theta}{2}\right) \]

### Final Answer:
\[ \frac{dy}{dx} = -\tan\left(\frac{3\theta}{2}\right) \]