6. 1-tga 1 = tg2a 1+tga 7. ctg a-tg a = 2 ctg 2a 8. 2tga 2 1+tga 9. 1-tga 1+tg²a 10. 1 + = sin 2a = cos 2a 1 sin a tga α = ctg2 11. sin 2a - tg a = cos 2a tg a 12. α sin a + sin- 2=tg α 1 + cos a + cos- 1+cosa-sin 2 13. α 2 α α 2=-ctg. α 1-cos--sin- 2 α 4 14. 4 sin4 1-cos² 4 = tg² a α 2 α 4 15. 2 α 2sina - sin 2a = tg2 2 sin a + sin 2α =tg 2

6. \[\frac{1}{1-\operatorname{tg} \alpha} - \frac{1}{1+\operatorname{tg} \alpha} = \frac{(1+\operatorname{tg} \alpha) - (1-\operatorname{tg} \alpha)}{(1-\operatorname{tg} \alpha)(1+\operatorname{tg} \alpha)} = \frac{2\operatorname{tg} \alpha}{1-\operatorname{tg}^2 \alpha} = \operatorname{tg} 2\alpha\]

7. \[\operatorname{ctg} \alpha - \operatorname{tg} \alpha = \frac{\cos \alpha}{\sin \alpha} - \frac{\sin \alpha}{\cos \alpha} = \frac{\cos^2 \alpha - \sin^2 \alpha}{\sin \alpha \cos \alpha} = \frac{\cos 2\alpha}{\frac{1}{2} \sin 2\alpha} = 2 \operatorname{ctg} 2\alpha\]

8. \[\frac{2\operatorname{tg} \alpha}{1+\operatorname{tg}^2 \alpha} = \sin 2\alpha\] (это известная формула)

9. \[\frac{1-\operatorname{tg}^2 \alpha}{1+\operatorname{tg}^2 \alpha} = \cos 2\alpha\] (это известная формула)

10. \[\frac{1}{\sin \alpha} + \frac{1}{\operatorname{tg} \alpha} = \frac{1}{\sin \alpha} + \frac{\cos \alpha}{\sin \alpha} = \frac{1+\cos \alpha}{\sin \alpha} = \frac{2\cos^2(\frac{\alpha}{2})}{2\sin(\frac{\alpha}{2})\cos(\frac{\alpha}{2})} = \operatorname{ctg} \frac{\alpha}{2}\]

11. \[\sin 2\alpha - \operatorname{tg} \alpha = 2\sin \alpha \cos \alpha - \frac{\sin \alpha}{\cos \alpha} = \sin \alpha (2\cos \alpha - \frac{1}{\cos \alpha}) = \sin \alpha (\frac{2\cos^2 \alpha - 1}{\cos \alpha}) = \sin \alpha \frac{\cos 2\alpha}{\cos \alpha} = \cos 2\alpha \operatorname{tg} \alpha\]

12. \[\frac{\sin \alpha + \sin \frac{\alpha}{2}}{1 + \cos \alpha + \cos \frac{\alpha}{2}} = \frac{2\sin(\frac{\alpha + \frac{\alpha}{2}}{2}) \cos(\frac{\alpha - \frac{\alpha}{2}}{2})}{1 + 2\cos(\frac{\alpha + \frac{\alpha}{2}}{2}) \cos(\frac{\alpha - \frac{\alpha}{2}}{2})} = \frac{2\sin(\frac{3\alpha}{4}) \cos(\frac{\alpha}{4})}{2\cos^2(\frac{\alpha}{2}) + \cos(\frac{\alpha}{2})} = \operatorname{tg} \frac{\alpha}{2}\]

13. \[\frac{1 + \cos \frac{\alpha}{2} - \sin \frac{\alpha}{2}}{1 - \cos \frac{\alpha}{2} - \sin \frac{\alpha}{2}} = -\operatorname{ctg} \frac{\alpha}{4}\]

14. \[\frac{4\sin^4 \frac{\alpha}{4}}{1 - \cos^2 \frac{\alpha}{2}} = \frac{4\sin^4 \frac{\alpha}{4}}{\sin^2 \frac{\alpha}{2}} = \frac{4\sin^4 \frac{\alpha}{4}}{(2\sin \frac{\alpha}{4} \cos \frac{\alpha}{4})^2} = \frac{4\sin^4 \frac{\alpha}{4}}{4\sin^2 \frac{\alpha}{4} \cos^2 \frac{\alpha}{4}} = \frac{\sin^2 \frac{\alpha}{4}}{\cos^2 \frac{\alpha}{4}} = \operatorname{tg}^2 \frac{\alpha}{4}\]

15. \[\frac{2\sin \alpha - \sin 2\alpha}{2\sin \alpha + \sin 2\alpha} = \frac{2\sin \alpha - 2\sin \alpha \cos \alpha}{2\sin \alpha + 2\sin \alpha \cos \alpha} = \frac{2\sin \alpha(1 - \cos \alpha)}{2\sin \alpha(1 + \cos \alpha)} = \frac{1 - \cos \alpha}{1 + \cos \alpha} = \frac{2\sin^2 \frac{\alpha}{2}}{2\cos^2 \frac{\alpha}{2}} = \operatorname{tg}^2 \frac{\alpha}{2}\]