The value of $$\frac{8}{\pi} \int_0^{\pi/2} \frac{\cos x^{2023}}{\sin x^{2023} + \cos x^{2023}} dx$$ is _____.

We need to evaluate $$\frac{8}{\pi} \int_0^{\pi/2} \frac{\cos^{2023}x}{\sin^{2023}x + \cos^{2023}x} \, dx$$.

Let:

$$I = \int_0^{\pi/2} \frac{\cos^{2023}x}{\sin^{2023}x + \cos^{2023}x} \, dx$$

King's property states that $$\int_0^a f(x)\,dx = \int_0^a f(a-x)\,dx$$. Applying this with $$a = \pi/2$$:

$$I = \int_0^{\pi/2} \frac{\cos^{2023}(\pi/2 - x)}{\sin^{2023}(\pi/2 - x) + \cos^{2023}(\pi/2 - x)} \, dx$$

Using $$\cos(\pi/2 - x) = \sin x$$ and $$\sin(\pi/2 - x) = \cos x$$:

$$I = \int_0^{\pi/2} \frac{\sin^{2023}x}{\cos^{2023}x + \sin^{2023}x} \, dx$$ $$I + I = \int_0^{\pi/2} \frac{\cos^{2023}x}{\sin^{2023}x + \cos^{2023}x} \, dx + \int_0^{\pi/2} \frac{\sin^{2023}x}{\cos^{2023}x + \sin^{2023}x} \, dx$$ $$2I = \int_0^{\pi/2} \frac{\cos^{2023}x + \sin^{2023}x}{\sin^{2023}x + \cos^{2023}x} \, dx = \int_0^{\pi/2} 1 \, dx = \frac{\pi}{2}$$ $$2I = \frac{\pi}{2}$$ $$I = \frac{\pi}{4}$$ $$\frac{8}{\pi} \times I = \frac{8}{\pi} \times \frac{\pi}{4} = \frac{8}{4} = 2$$

The answer is 2.