At which point does the curve x=1-2(cos^2)t, y=(tant)(1-2(cos^2)t) cross itself?

Question

At what point does the curve x=1-2(cos^2)t, y=(tant)(1-2(cos^2)t) cross itself? Find the equations of both tangents at that point.

How do you know?

Answer 1

At which point does the curve
x = 1 - 2cos²t; y= (tan t)(1 - 2cos²t)
cross itself?
________

It crosses itself at the origin.

Plug in the values t = π/4, 3π/4.

t = π/4 → (x, y) = (0, 0)
t = 3π/4 → (x, y) = (0, 0)
________

dx/dt = 4(cos t)(sin t)

dy/dt = (sec²t)(1 - 2cos²t) + (tan t)[4(cos t)(sin t)]
dy/dt = sec²t - 2 + 4sin²t

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

t = π/4
dy/dx = (2 - 2 + 4/2) / 2 = 2/2 = 1

t = 3π/4
dy/dx = (2 - 2 + 4/2) / -2 = 2/-2 = -1

The equations of the tangent lines at the origin are:

y = x
y = -x

Answer 2

Certainly crosses at (0,0) when t = pi/4 or -pi/4. Now you can compute dy/dt and divide that by dx/dt, then evaluate at those t values o find the proper slopes..tangent line equations are of the from y = mx since they pass through the origin.

Answer 3

It doesn't.