Задания для самостоятельной работы Упростите выражения: 1. $$1 - \cos^2a$$; 2. $$\sin^2a -1$$; 3. $$(1-\cos a)(1 + \cos a)$$; 4. $$\sin a \cdot tg a \cdot ctg a$$; 5. $$tg a \cdot ctg a + ctg^2 a$$; 6. $$(\sin a + \cos a)^2 - 2\sin a \cdot \cos a$$; 7. $$\frac{\sin^4 a + 2\sin^2 a \cdot \cos^2 a + \cos^4 a}{\sin^4 a - \cos^4 a}$$; 8. $$\frac{\sin^2 a - \cos^2 a}{\cos a}$$; 9. Докажите тождество: $$(1-\sin^2 a)(1 + tg^2 a) = 1$$; $$\sin^2 a (1 + ctg^2 a) - \cos^2 a = \sin^2 a$$; $$\cos^4 a - \sin^4 a = \cos^2 a - \sin^2 a$$;

1. $$1 - \cos^2 a = \sin^2 a$$ 2. $$\sin^2 a - 1 = -\cos^2 a$$ 3. $$(1-\cos a)(1 + \cos a) = 1 - \cos^2 a = \sin^2 a$$ 4. $$\sin a \cdot tg a \cdot ctg a = \sin a \cdot \frac{\sin a}{\cos a} \cdot \frac{\cos a}{\sin a} = \sin a$$ 5. $$tg a \cdot ctg a + ctg^2 a = 1 + ctg^2 a = \frac{1}{\sin^2 a}$$ 6. $$(\sin a + \cos a)^2 - 2\sin a \cdot \cos a = \sin^2 a + 2\sin a \cos a + \cos^2 a - 2\sin a \cos a = \sin^2 a + \cos^2 a = 1$$ 7. $$\frac{\sin^4 a + 2\sin^2 a \cdot \cos^2 a + \cos^4 a}{\sin^4 a - \cos^4 a} = \frac{(\sin^2 a + \cos^2 a)^2}{(\sin^2 a - \cos^2 a)(\sin^2 a + \cos^2 a)} = \frac{1}{\sin^2 a - \cos^2 a} = -\frac{1}{\cos^2 a - \sin^2 a}$$ 8. $$\frac{\sin^2 a - \cos^2 a}{\cos a} = \frac{-(\cos^2 a - \sin^2 a)}{\cos a} = -\frac{\cos 2a}{\cos a}$$ 9. Докажем тождество: * $$(1-\sin^2 a)(1 + tg^2 a) = 1$$ $$\cos^2 a \cdot (1 + \frac{\sin^2 a}{\cos^2 a}) = \cos^2 a + \sin^2 a = 1$$ * $$\sin^2 a (1 + ctg^2 a) - \cos^2 a = \sin^2 a$$ $$\sin^2 a \cdot (1 + \frac{\cos^2 a}{\sin^2 a}) - \cos^2 a = \sin^2 a + \cos^2 a - \cos^2 a = \sin^2 a$$ * $$\cos^4 a - \sin^4 a = \cos^2 a - \sin^2 a$$ $$\cos^4 a - \sin^4 a = (\cos^2 a - \sin^2 a)(\cos^2 a + \sin^2 a) = (\cos^2 a - \sin^2 a) \cdot 1 = \cos^2 a - \sin^2 a$$