Let $\int \frac{1}{1-\mathrm{sinx}}\sqrt{\frac{\mathrm{cosx}}{1+\mathrm{cosx}+\mathrm{sinx}}}\mathrm{dx}=\frac{2\sqrt{2}}{\sqrt{\mathrm{f}\left(\mathrm{x}\right)}}+\mathrm{C},$ when $\mathrm{x}\in \left(-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right)$ and C is the constant of integration, then

Given integral $=\int \frac{1}{{\left(\cos \frac{\mathrm{x}}{2}-\sin \frac{\mathrm{x}}{2}\right)}^2}\sqrt{\frac{{\cos }^2\frac{\mathrm{x}}{2}-{\sin }^2\frac{\mathrm{x}}{2}}{2{\cos }^2\frac{\mathrm{x}}{2}+2\cos \frac{\mathrm{x}}{2}\sin \frac{\mathrm{x}}{2}}}\mathrm{dx}$
$=\int \frac{\mathrm{dx}}{{\left(\cos \frac{\mathrm{x}}{2}-\sin \frac{\mathrm{x}}{2}\right)}^{3/2}\sqrt{2\cos \frac{\mathrm{x}}{2}}}=\frac{-2}{\sqrt{2}}\int \frac{-\frac{1}{2}{\sec }^2\frac{\mathrm{x}}{2}\mathrm{dx}}{{\left(1-\tan \frac{\mathrm{x}}{2}\right)}^{3/2}}$
$=-\sqrt{2}\frac{{\left(1-\tan \frac{\mathrm{x}}{2}\right)}^{-1/2}}{-\frac{1}{2}}+\mathrm{C}$
$\therefore \mathrm{f}\left(\mathrm{x}\right)=1-\tan \frac{\mathrm{x}}{2}$