Let us consider the parametric curve $f(t)=(\cos^3t \cos(3t), \cos^3t \sin(3t))$, where $t$ belongs to $\mathbb R$. Then show that the curve is closed with exactly one self-intersection.
I know that a closed curve means if the curve is periodic for some period $T≠0$ and not constant. And the curve $f$ is said to have a self-intersection at a point $p$ of the curve if there exist parameter values $a≠b$ such that $f(a)=f(b)=p$ and if $f$ is closed with period $T$, then $a-b$ is not an integer multiple of $T$.
I have tried to show the closedness condition but the calculation become very large.
Like: let $T$ be the period of $f$, then $f(t+T)=f(t)$ which implies two equations those are $\cos^3(t+T)\cos3(t+T)=\cos^3t\cos(3t)$ and $\cos^3(t+T) \sin(3(t+T))=\cos^3t\sin(3t)$ but after a huge calculation failed to find the value of $T$ i.e. period.
Please help me to solve this.