For the curve y = 4x³ – 2x⁵, find all the points at which the tangent passes through the origin. 

The equation of the given curve is y = 4x³ − 2x⁵.

⇒ 8cosθ + 4 = 4 + cos²θ + 4 cosθ 

Therefore, the slope of the tangent at a point (x, y) is 12x² − 10x⁴.

The equation of the tangent at (x, y) is given by,

⇒ cos²θ  - 4 cosθ  = 0…………(1)

When the tangent passes through the origin (0, 0), then X = Y = 0.

Therefore, equation (1) reduces to:

⇒ cosθ  (cosθ  - 4) = 0

⇒ cosθ  = 0 or cosθ  = 4