Q5:

If $x$ and $y$ are connected parametrically by the equations, without eliminating the parameter, Find $\frac{dy}{dx}$. $x = \cos \theta - \cos 2\theta, y = \sin \theta - \sin 2\theta$

Step 1: Identifying the Parametric Equations

The variables $x$ and $y$ are functions of the parameter $\theta$:

• $x = \cos \theta - \cos 2\theta$
• $y = \sin \theta - \sin 2\theta$

To find $\frac{dy}{dx}$, we apply the Chain Rule for parametric differentiation: $\frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta}$ [Per the Chain Rule for Parametric Differentiation].

Step 2: Differentiating $x$ with respect to $\theta$

Differentiating $x = \cos \theta - \cos 2\theta$ with respect to $\theta$:

$\frac{dx}{d\theta} = \frac{d}{d\theta}(\cos \theta) - \frac{d}{d\theta}(\cos 2\theta)$

Applying the standard derivative $\frac{d}{d\theta}(\cos \theta) = -\sin \theta$ and the Chain Rule $\frac{d}{d\theta}(\cos 2\theta) = -\sin 2\theta \cdot \frac{d}{d\theta}(2\theta) = -2\sin 2\theta$ [By Rule of Chain Rule]:

$\frac{dx}{d\theta} = -\sin \theta - (-2 \sin 2\theta)$

$\frac{dx}{d\theta} = 2 \sin 2\theta - \sin \theta$

Step 3: Differentiating $y$ with respect to $\theta$

Differentiating $y = \sin \theta - \sin 2\theta$ with respect to $\theta$:

$\frac{dy}{d\theta} = \frac{d}{d\theta}(\sin \theta) - \frac{d}{d\theta}(\sin 2\theta)$

Applying the standard derivative $\frac{d}{d\theta}(\sin \theta) = \cos \theta$ and the Chain Rule $\frac{d}{d\theta}(\sin 2\theta) = \cos 2\theta \cdot 2$:

$\frac{dy}{d\theta} = \cos \theta - 2 \cos 2\theta$

Step 4: Calculating $\frac{dy}{dx}$

Substitute the results from Step 2 and Step 3 into the quotient expression:

$\frac{dy}{dx} = \frac{\cos \theta - 2 \cos 2\theta}{2 \sin 2\theta - \sin \theta}$

Visual Representation of Parametric Slope

Step 5: Final Simplification (Optional)

While the expression is complete, it is mathematically standard to present the result in its derived form as further trigonometric expansion using double-angle identities ($\cos 2\theta = 2\cos^2\theta - 1$ and $\sin 2\theta = 2\sin\theta\cos\theta$) does not lead to significant further reduction in this specific rational function.

Final Solution: $\frac{dy}{dx} = \frac{\cos \theta - 2 \cos 2\theta}{2 \sin 2\theta - \sin \theta}$