13.9.° Упростите выражение: 1) $\frac{\sin 2\alpha}{\sin \alpha}$; 2) $\frac{\sin 2\alpha}{\cos^2 \alpha - \sin^2 \alpha}$; 3) $\cos 2\alpha + \sin^2 \alpha$; 4) $\frac{\sin 50^\circ}{2\cos 25^\circ}$; 5) $\frac{\cos 2\alpha}{\cos \alpha - \sin \alpha}$; 6) $1 - 2\sin^2 \frac{\alpha}{4}$; 7) $(\sin \frac{\alpha}{4} + \cos \frac{\alpha}{4})(\sin \frac{\alpha}{4} - \cos \frac{\alpha}{4})$; 8) $\frac{\sin \alpha \cos \alpha}{1 - 2 \sin^2 \alpha}$

1) $\frac{\sin 2\alpha}{\sin \alpha} = \frac{2\sin \alpha \cos \alpha}{\sin \alpha} = 2\cos \alpha$ 2) $\frac{\sin 2\alpha}{\cos^2 \alpha - \sin^2 \alpha} = \frac{2\sin \alpha \cos \alpha}{\cos^2 \alpha - \sin^2 \alpha} = \frac{2\sin \alpha \cos \alpha}{\cos 2\alpha}$ - Без дополнительной информации это выражение не упрощается. 3) $\cos 2\alpha + \sin^2 \alpha = \cos^2 \alpha - \sin^2 \alpha + \sin^2 \alpha = \cos^2 \alpha$ 4) $\frac{\sin 50^\circ}{2\cos 25^\circ} = \frac{\sin (2 \cdot 25^\circ)}{2\cos 25^\circ} = \frac{2\sin 25^\circ \cos 25^\circ}{2\cos 25^\circ} = \sin 25^\circ$ 5) $\frac{\cos 2\alpha}{\cos \alpha - \sin \alpha} = \frac{\cos^2 \alpha - \sin^2 \alpha}{\cos \alpha - \sin \alpha} = \frac{(\cos \alpha - \sin \alpha)(\cos \alpha + \sin \alpha)}{\cos \alpha - \sin \alpha} = \cos \alpha + \sin \alpha$ 6) $1 - 2\sin^2 \frac{\alpha}{4} = \cos (2 \cdot \frac{\alpha}{4}) = \cos \frac{\alpha}{2}$ 7) $(\sin \frac{\alpha}{4} + \cos \frac{\alpha}{4})(\sin \frac{\alpha}{4} - \cos \frac{\alpha}{4}) = \sin^2 \frac{\alpha}{4} - \cos^2 \frac{\alpha}{4} = -\cos (2 \cdot \frac{\alpha}{4}) = -\cos \frac{\alpha}{2}$ 8) $\frac{\sin \alpha \cos \alpha}{1 - 2 \sin^2 \alpha} = \frac{\frac{1}{2} \sin 2\alpha}{\cos 2\alpha} = \frac{1}{2} \tan 2\alpha$