Номер 510

Доказать тождество:

1. $\dfrac{\cos2\alpha}{\sin\alpha\cos\alpha+\sin^2\alpha}=\operatorname{ctg}\alpha-1$;
2. $\dfrac{\sin2\alpha-2\cos\alpha}{\sin\alpha-\sin^2\alpha}=-2\operatorname{ctg}\alpha$;
3. $\operatorname{tg}\alpha(1+\cos2\alpha)=\sin2\alpha$;
4. $\dfrac{1-\cos2\alpha+\sin2\alpha}{1+\cos2\alpha+\sin2\alpha}\cdot\operatorname{ctg}\alpha=1$;
5. $\dfrac{(1-2\cos^2\alpha)(2\sin^2\alpha-1)}{4\sin^2\alpha\cos^2\alpha}=\operatorname{ctg}^22\alpha$;
6. $1-2\sin^2\left(\frac{\pi}{4}-\frac{\alpha}{2}\right)=\sin\alpha$;
7. $\dfrac{\sin\alpha+\sin2\alpha}{1+\cos\alpha+\cos2\alpha}=\operatorname{tg}\alpha$.