Evaluate $\int_0^\frac{\pi}{2} \frac{\cos^{9999}(x)}{\cos^{9999}(x)+sin^{9999}(x)} dx$

Question

$\int_0^\frac{\pi}{2} \frac{\cos^{9999}(x)}{\cos^{9999}(x)+sin^{9999}(x)} dx$

Using desmos the value of definte integral is around $\frac{\pi}{4}$ but I don't know where to start.