At what point does the curve? (parametric)?

Question

at what point does the curve x=1-2(cost)^2, y=(tan(t))(1-2(cost)^2)

cross itself? find the equations of both tangents at that point?

if would help if someone could help me find out the method i should use to derive such a problem

Answer 1

let x=y

you'll get 1=tant

Hence, t must be 45degrees

When t=45 degrees; x=1-1=0 and y=0

So, they intersect at origin

Now dx/dt=4costsint

So, slope of the first curve at origin is 0 and hence, the slope of the

second curve is tan 45 degrees=1

Hence, the equations of the tangents are x=0 and y=t

Answer 2

FYI, one does not derive a problem, but solve it.

A curve crosses itself means for 2 different values of t there is same x and same y value.

So find u and v such thqt u=/= v and

1-2(cosu)^2 = 1-2(cosv)^2, and

(tan(u))(1-2(cosu)^2) = (tan(v))(1-2(cosv)^2) . Now 1-2(cost)^2 = 0 cannot be sol as tant will not be defined. So tant = 0 or t =0 and pi works.

Now find dy/dx for these 2 and write the eqn of the tgt.