Solving $\frac{\cos^2\left(\frac\pi2 \cos\theta\right)}{\sin^2\theta} = 0.5$ 
$$\frac{\cos^2\left(\dfrac\pi2 \cos\theta\right)}{\sin^2\theta} = 0.5$$

I want to solve the above equation for $\theta$ in order to find its value, but I am stuck.
Could anyone enlighten me by a method to solve it?

Let $x =\cos\theta$ to simplify the equation to
$$\cos(\pi x) +x^2=0$$
which has the trivial roots $\pm 1$ (excluded due to $\sin\theta \ne 0$), as well as the root that can be approximated with $\frac\pi2-\pi x+x^2 =0$, i.e.
$$x=\left(1+\sqrt{1-\frac2\pi} \right)^{-1}= 0.6239$$
(vs. the exact $ 0.6298$). Thus, the solutions are
$$\theta = 2\pi k\pm \text{arcsec} \left(1+\sqrt{1-\frac2\pi} \right) $$
