Question

if x= 2 cos theta - cos 2 theta and y=2sin theta - sin 2theta then prove that dy/dx = tan (3 theta/2)

Answer 1

Hope you have understood

Answer 2

Answer:

$x=2 cos\theta - cos 2\theta$ -------(1)

And,

$y=2sin\theta - sin 2\theta$   -----(2),

By differentiating equation (1) and (2) with respect to $\theta$,

We get,

$\frac{dx}{d\theta}=-2 sin\theta + 2 sin 2\theta$ ------(3)

$\frac{dy}{d\theta}=2 cos\theta - 2 cos 2\theta$ ------(4)

On dividing equation (4) by equation (3),

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

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

$=\frac{2 sin(\frac{\theta+2\theta}{2}) sin (\frac{2\theta-\theta}{2})} {2 cos(\frac{\theta+2\theta}{2})sin (\frac{2\theta-\theta}{2})} }$

$=\frac{2 sin\frac{3\theta}{2}\times sin \theta}{2 cos \frac{3\theta}{2}\times sin \theta}$

$=\frac{sin \frac{3\theta}{2}}{cos \frac{3\theta}{2}}$

$=tan\frac{3\theta}{2}$

Hence, proved.