X = cos (2t) and y = sin (2t)

Find the second derivative

Dy/dx = (dy/dt) / (dx/dt)

dy/dt = 2cos (2t)

dx/dt = - 2sin (2t)

dy/dx = 2cos (2t) / - 2sin (2t) = - cot (2t)

d^y/dx^2 = d (-1/tan (2t)) / dx = (tan (2t) * 0 - (-1) * 2sec^2 (2t)) / tan^2 (2t) = 2sec^2 (2t) / tan^2 (2t) = 2csc^2 (2t)