Counting the solutions of a parameter equation

You will count the solutions of $\left(\sin^{-1}x\right)^2 + \left(\cos^{-1}x\right)^2 = a$ for any value of $a$, by the method of the lesson. The checker compares your answers with a count made by scanning the numbers $x$ on a fine grid.

The idea. Put $t = \sin^{-1}x$. The left side becomes $t^2 + \left(\dfrac{\pi}{2} - t\right)^2$. The equation is a quadratic in $t$ with the roots $t = \dfrac{\pi}{4} \pm s$, where $s = \sqrt{\dfrac{a - \pi^2/8}{2}}$. Only the roots in $\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$ give a solution $x$.

Your tasks.

1. Finish g(t). Return $t^2 + \left(\dfrac{\pi}{2} - t\right)^2$.
2. Finish roots_in_branch(a). Return the sorted list of the roots $t$ that lie in $\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$. Return an empty list if $a$ is below $\dfrac{\pi^2}{8}$.
3. Finish count_x(a). Return how many solutions $x$ there are.