$$
\int_0^\pi \frac{(\cos x)^2\,\mathrm{d}x}{1 + \cos x \sin x}
= \int_0^\pi \frac{(\sin x)^2\,\mathrm{d}x}{1 - \cos x \sin x}
= \int_0^\pi \frac{(\cos x)^2\,\mathrm{d}x}{1 + \cos x \sin x}
= \int_0^\pi \frac{(\sin x)^2\,\mathrm{d}x}{1 - \cos x \sin x}
$$