Place where parametric curve itself itself

[motornoob101]

Homework Statement

Find the place where the parametric curve intersect itself

x = 1-2cos^{2}t
y = tant(1-2cos^{2}t)

Homework Equations

The Attempt at a Solution

So I started with the x values..

1-2cos^{2}t_{1} = 1-2cos^{2}t_{2}

By canceling the same stuff on both sides, I got
cos^{2}t_{1} = cos^{2}t_{2}

Then I tried with y
I rewrote y in a different form.

y_{1} = tan t (x_{1})
and
y_{2} = tan t(x_{2})

This is possible since y already contains an expression for x.

Since the curve intersect itself, we know x1 must equal x2 so they cancel out.

then I am left with tant_{1}= tant_{2}

but I can't solve for t

Appreciate any help. Thanks

 

[HallsofIvy]

Why can't you? Since tangent is periodic with period \pi, but one-to-one within each period, tan(t₁)= tan(t₂) requires that t_1= t_2+ n\pi. Now, which of those values satifies cos²(t₁)= cos²(t₂)?

 

[motornoob101]

Ah, now I know what you meant, thanks!