Question

Упростить. Помогите, пожалуйста!

Answer 1

$tg(\frac{\pi}{4}-\frac{x}{2})=\frac{tg\frac{\pi}{4}-tg\frac{x}{2}}{1+tg\frac{\pi}{4}\cdot tg\frac{x}{2}}=\frac{1-tg\frac{x}{2}}{1+tg\frac{x}{2}}=\frac{cos\frac{x}{2}-sin\frac{x}{2}}{cos\frac{x}{2}+sin\frac{x}{2}}=\frac{(cos\frac{x}{2}-sin\frac{x}{2})^2}{cos^2\frac{x}{2}-sin^2\frac{x}{2}}=\\\\=\frac{cos^2\frac{x}{2}-2sin\frac{x}{2}\cdot cos\frac{x}{2}+sin^2\frac{x}{2}}{cos(2\cdot \frac{x}{2})}=\frac{1-sin(2\frac{x}{2})}{cosx}=\frac{1-sinx}{cosx}\\\\\\\frac{sinx}{tg(\frac{\pi}{4}-\frac{x}{2})(1+sinx)}=$

$=\frac{sinx}{\frac{1-sinx}{cosx}(1+sinx)}=\frac{sinx*cosx}{1-sin^2x}=\frac{sinx*cosx}{cos^2x}=\frac{sinx}{cosx}=tgx$