If `x=2costheta-cos2theta,y=2sintheta-sin2theta,"find "dy/dx" at "theta=pi/2`.

Answer

To find \( \frac{dy}{dx} \) at \( \theta = \frac{\pi}{2} \), we will use the chain rule, which states that:

\[ \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} \]

### Step 1: Define the functions
We are given:
\[ x = 2 \cos \theta - \cos 2\theta \]
\[ y = 2 \sin \theta - \sin 2\theta \]

### Step 2: Differentiate \( y \) with respect to \( \theta \)
To find \( \frac{dy}{d\theta} \), we differentiate \( y \):
\[ \frac{dy}{d\theta} = \frac{d}{d\theta}(2 \sin \theta - \sin 2\theta) \]
Using the derivatives:
- The derivative of \( \sin \theta \) is \( \cos \theta \).
- The derivative of \( \sin 2\theta \) is \( 2 \cos 2\theta \) (using the chain rule).

Thus,
\[ \frac{dy}{d\theta} = 2 \cos \theta - 2 \cos 2\theta \]

### Step 3: Differentiate \( x \) with respect to \( \theta \)
Now we differentiate \( x \):
\[ \frac{dx}{d\theta} = \frac{d}{d\theta}(2 \cos \theta - \cos 2\theta) \]
Using the derivatives:
- The derivative of \( \cos \theta \) is \( -\sin \theta \).
- The derivative of \( \cos 2\theta \) is \( -2 \sin 2\theta \) (using the chain rule).

Thus,
\[ \frac{dx}{d\theta} = -2 \sin \theta + 2 \sin 2\theta \]

### Step 4: Evaluate \( \frac{dy}{d\theta} \) and \( \frac{dx}{d\theta} \) at \( \theta = \frac{\pi}{2} \)
Now we substitute \( \theta = \frac{\pi}{2} \):
1. For \( \frac{dy}{d\theta} \):
\[ \frac{dy}{d\theta} = 2 \cos\left(\frac{\pi}{2}\right) - 2 \cos\left(2 \cdot \frac{\pi}{2}\right) = 2 \cdot 0 - 2 \cdot (-1) = 0 + 2 = 2 \]

2. For \( \frac{dx}{d\theta} \):
\[ \frac{dx}{d\theta} = -2 \sin\left(\frac{\pi}{2}\right) + 2 \sin\left(2 \cdot \frac{\pi}{2}\right) = -2 \cdot 1 + 2 \cdot 0 = -2 + 0 = -2 \]

### Step 5: Calculate \( \frac{dy}{dx} \)
Now we can find \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{2}{-2} = -1 \]

### Final Answer
Thus, the value of \( \frac{dy}{dx} \) at \( \theta = \frac{\pi}{2} \) is:
\[ \frac{dy}{dx} = -1 \]

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