`x= 2 cos t -cos 2t ,y=2 sin t-sin 2t ` Find d y / d x

Answer

To find \(\frac{dy}{dx}\) for the given parametric equations \(x = 2 \cos t - \cos 2t\) and \(y = 2 \sin t - \sin 2t\), we will follow these steps:

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

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

We differentiate \(x\) with respect to \(t\):
\[ \frac{dx}{dt} = \frac{d}{dt}(2 \cos t) - \frac{d}{dt}(\cos 2t) \]
Using the derivative of \(\cos x\) which is \(-\sin x\), we have:
\[ \frac{dx}{dt} = 2(-\sin t) - (-\sin 2t \cdot 2) = -2 \sin t + 2 \sin 2t \]
Thus,
\[ \frac{dx}{dt} = -2 \sin t + 2 \sin 2t \]

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

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

We differentiate \(y\) with respect to \(t\):
\[ \frac{dy}{dt} = \frac{d}{dt}(2 \sin t) - \frac{d}{dt}(\sin 2t) \]
Using the derivative of \(\sin x\) which is \(\cos x\), we have:
\[ \frac{dy}{dt} = 2 \cos t - (\cos 2t \cdot 2) = 2 \cos t - 2 \cos 2t \]
Thus,
\[ \frac{dy}{dt} = 2 \cos t - 2 \cos 2t \]

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

Using the chain rule, we find:
\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \]
Substituting the values we found:
\[ \frac{dy}{dx} = \frac{2 \cos t - 2 \cos 2t}{-2 \sin t + 2 \sin 2t} \]
We can simplify this by factoring out the common factor of \(2\):
\[ \frac{dy}{dx} = \frac{2(\cos t - \cos 2t)}{2(-\sin t + \sin 2t)} = \frac{\cos t - \cos 2t}{-\sin t + \sin 2t} \]

### Step 4: Simplify using 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
\[ \sin a - \sin b = 2 \sin\left(\frac{a-b}{2}\right) \cos\left(\frac{a+b}{2}\right) \]
we can rewrite:
\[ \frac{dy}{dx} = \frac{-2 \sin\left(\frac{t + 2t}{2}\right) \sin\left(\frac{t - 2t}{2}\right)}{2 \sin\left(\frac{2t - t}{2}\right) \cos\left(\frac{2t + t}{2}\right)} \]
This simplifies to:
\[ \frac{dy}{dx} = \frac{-\sin\left(\frac{3t}{2}\right) \sin\left(-\frac{t}{2}\right)}{\sin\left(\frac{t}{2}\right) \cos\left(\frac{3t}{2}\right)} \]
Using \(\sin(-x) = -\sin(x)\):
\[ \frac{dy}{dx} = \frac{\sin\left(\frac{3t}{2}\right) \sin\left(\frac{t}{2}\right)}{\sin\left(\frac{t}{2}\right) \cos\left(\frac{3t}{2}\right)} \]
Cancelling \(\sin\left(\frac{t}{2}\right)\):
\[ \frac{dy}{dx} = \frac{\sin\left(\frac{3t}{2}\right)}{\cos\left(\frac{3t}{2}\right)} = \tan\left(\frac{3t}{2}\right) \]

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