Упр.507 ГДЗ Алимов 10-11 класс (Алгебра)

1. $$\frac{\sin 2a}{(\sin a+\cos a)^2-1}$$
2. $$\frac{1+\cos 2a}{1-\cos 2a}$$

1) $$\frac{\sin 2\alpha}{(\sin \alpha+\cos \alpha)^2-1}=\frac{\sin 2\alpha}{\sin^2\alpha+2\sin\alpha\cos\alpha+\cos^2\alpha-1}$$

$$=\frac{\sin 2\alpha}{1+\sin 2\alpha-1}=\frac{\sin 2\alpha}{\sin 2\alpha}=1.$$

2) $$\frac{1+\cos 2\alpha}{1-\cos 2\alpha}=\frac{1+(\cos^2\alpha-\sin^2\alpha)}{1-(\cos^2\alpha-\sin^2\alpha)}$$

$$=\frac{1-\sin^2\alpha+\cos^2\alpha}{1-\cos^2\alpha+\sin^2\alpha}=\frac{\cos^2\alpha+\cos^2\alpha}{\sin^2\alpha+\sin^2\alpha}=\frac{2\cos^2\alpha}{2\sin^2\alpha}=\left(\frac{\cos\alpha}{\sin\alpha}\right)^2=\operatorname{ctg}^2\alpha.$$

Ответ: 1) $$1$$; 2) $$\operatorname{ctg}^2\alpha$$.