if ` x =2 cos theta - cos 2 theta `
` y = 2 sin theta - sin 2 theta`
Find ` (d^(2)y)/(dx^(2)) `at `theta = pi/2`

Explanation

` (dx)/(dtheta) = 2 sin theta = 2 sin 2 theta`
` (dy)/(dtheta) = 2 cos theta - 2 cos 2 theta`
` Rightarrow (dy)/(dx) = ((dy)/(dtheta))/((dx)/(d theta) = (2 cos theta - 2 cos 2 theta ) / (-2 sin theta - sin 2 theta)`
`= (cos theta - cos 2 theta)/(sin 2 theta - sin theta ) `
` ( 2 sin ""(3theta)/2 sin "" theta/2)/(2 cos ""(3theta)/2 sin ""theta/2) = tan"" (3theta)/2`
Differentiating w.r.t x, we get
` (d^(2)y)/(dx^(2)) = sec^(2)"" (3 theta)/2 . 3/2 (dtheta )/(dx)`
` 3/2 1/(cos^(2) ""((3 theta)/2)) . 1/(-2 sin theta + 2 sin 2 theta ) `
` 3/8 (1/(cos^(3)) "" (3 theta)/2 sin"" theta/2 ) `
`((d^(2)y)/(dx^(2)))_(theta=pi/2) = 3/(8 cos^(3) ((3pi)/4)sin""pi/4) = 3/(8(-1/(2sqrt2))1/sqrt2)`
` = - (3 xx4)/8`
`=- 3/2`