prove that a function is an immersion

Question

How I can show that $F \colon \mathbb{R} \to \mathbb{R}^2$ defined by $F(t)= (\cos (t), \sin(t))$ is an immersion? In my definition $F$ is an immersion if $\forall p$,$dF_p$ is injective. I have compute $dF_p=(-\sin (t), \cos(t))$. But now?

Answer 1

Claim: The linear map $dF_t:\mathbb{R}\rightarrow\mathbb{R}^2$, $x\mapsto (-\sin(t)x,\cos(t)x)$ is injective.

Proof: Since if $dF_t(x)=0$ for $x\not=0$, then $-\sin(t)x=0$ and $\cos(t)x=0$. But then $\cos(t)=\sin(t)=0$.

$\sin$ and $\cos$ have no common zeros. Therefore $\ker dF_t = 0$.$\square$

In other words, the matrix $(-\sin(t),\cos(t))$ always has rank $1$.

This implies that $F$ is an immersion.