Given the hyperbolic metric $ds^2=\frac{dx^2+dy^2}{x^2}$ on the half plane $x > 0$, find the length of the arc of the circle $x^2+y^2=1$ from $(\cos\alpha,\sin\alpha)$ to $(\cos \beta, \sin\beta)$
I found that $ds^2=\displaystyle\frac{d\theta^2}{\cos^2\theta}$ but when I try to plug in $\pi/3, -\pi/3$, which should give me the arc length of $2\pi/3$,
I get $4\pi/3=\sqrt{\displaystyle\frac{{(\pi/3-(-\pi/3))}^2}{cos^2{(\pi/3})}}$
I feel like I'm making a simple mistake but I cant place it