Find $\frac{d y}{d x}$ , if x and y are connected parametrically by the equations given in questions without eliminating the parameter.

$x = c o s θ - c o s 2 θ, y = s i n θ - s i n 2 θ .$

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Solution

Given, $x = c o s θ - c o s 2 θ, y = s i n θ - s i n 2 θ .$

Differentiating w.r.t. $θ$ , we get

$∴ \frac{d x}{d θ} = \frac{d}{d θ} (c o s θ - c o s 2 θ) = - s i n θ - (- s i n 2 θ) 2 = - s i n θ + 2 s i n 2 θ a n d \frac{d y}{d θ} = \frac{d}{d θ} (s i n θ - s i n 2 θ) = c o s θ - (c o s 2 θ) 2 = c o s θ - 2 c o s 2 θ \Rightarrow \frac{d y}{d x} = \frac{\frac{d y}{d θ}}{\frac{d x}{d θ}} = \frac{d y}{d θ} \times \frac{d θ}{d x} = \frac{c o s θ - 2 c o s 2 θ}{2 s i n 2 θ - s i n θ}$

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