Question about polar coordinates in relation to $x^2-y^2$

Question

I was reading about polar coordinates, and I know that $x^2 + y^2 = r^2$, however, what is the polar coordinate equivalent to $x^2 - y^2$?

Answer 1

Note that in polar cordinate, you have $$\left\{\begin{align*}x&:=&r\cos(\theta)\\ y&:=&r\sin(\theta)\end{align*}\right.$$where $r\in \mathbb{R}^{+}, \theta\in [0,2\pi)$.

Now, you have $$x=r\cos(\theta) \implies x^{2}=r^{2}\cos^{2}(\theta)$$and $$y=r\sin(\theta) \implies y^{2}=r^{2}\sin^{2}(\theta)$$Then, you have for example $$x^{2}-y^{2}=r^{2}(\cos^{2}(\theta)-\sin^{2}(\theta))\overbrace{=}^{\cos^{2}(\theta)+\sin^{2}(\theta)=1}=r^{2}(1-2\sin^{2}(\theta))\overbrace{=}^{1-2\sin^{2}(\theta)=\cos(2 \theta)}r^{2}\cos(2\theta).$$