If x = a(cos2t + 2tsin 2t) and y = a(sin2t - 2t cos 2t),then find d^2y/dx^2.

x = [{a(cos2t + 2t sin2t)}]
 dx/dt = a[-2 sin2t + 2t(2 cos2t) + 2 sin2t] = 4at cos2t
​Now, dy /dt = d/dt [a(sin2t - 2t cos2t)]
= a[2cos2t - { 2t(-2 sin2t) + 2 cos2t)}]
=4at sin2t
Now, dy/dx = 4at sin2t / 4at cos2t = tan2t
 d²y / dx² = 2 sec²(2t) . dt/dx  = ​2 sec²(2t) . sec 2t/4at