Show that the curve x = 6 cos (t), y = 5 sin (t) cos (t) has two tangents at (0, 0) and find their equations.

We can write

cos (t) = x/6

so

y = 5 (±√ (1 - (x/6) ^2) * x/6

Then

y' = (±5/6) * (√ (1 - (x/6) ^2) + x / (2√ (1 - (x/6) ^2) * (-2 (x/6))

The limit as x → 0 is ±5/6

The equations of the two tangents are

y = (5/6) x

y = (-5/6) x