Упр.510 ГДЗ Алимов 10-11 класс (Алгебра)

Доказать тождество:

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

   $$\frac{\cos a-\sin a}{\sin a}=\frac{\cos a}{\sin a}-1=\ctg a-1$$

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

3. $$\tg a(1+\cos 2a)=\tg a\bigl((\cos^2 a+\sin^2 a)+(\cos^2 a-\sin^2 a)\bigr)=\tg a\cdot 2\cos^2 a$$

   $$\tg a\cdot 2\cos^2 a=\frac{\sin a}{\cos a}\cdot 2\cos^2 a=2\sin a\cos a=\sin 2a$$

4. $$\frac{1-\cos 2a+\sin 2a}{1+\cos 2a+\sin 2a}\cdot \ctg a= \frac{(1-\cos 2a)+\sin 2a}{(1+\cos 2a)+\sin 2a}\cdot \ctg a$$

   $$=\frac{2\sin^2 a+2\sin a\cos a}{2\cos^2 a+2\sin a\cos a}\cdot \ctg a =\frac{2\sin a(\sin a+\cos a)}{2\cos a(\cos a+\sin a)}\cdot \ctg a$$

   $$=\frac{\sin a}{\cos a}\cdot \ctg a=\tg a\cdot \ctg a=1$$

5. $$\frac{(1-2\cos^2 a)(2\sin^2 a-1)}{4\sin^2 a\cos^2 a}= \frac{(\sin^2 a-\cos^2 a)(\sin^2 a-\cos^2 a)}{(2\sin a\cos a)^2}$$

   $$=\frac{\cos^2 2a}{\sin^2 2a}=\ctg^2 2a$$

6. $$1-2\sin^2\left(\frac{\pi}{4}-\frac{a}{2}\right)=\cos\left(2\left(\frac{\pi}{4}-\frac{a}{2}\right)\right)=\cos\left(\frac{\pi}{2}-a\right)=\sin a$$

7. $$\frac{\sin a+\sin 2a}{1+\cos a+\cos 2a} =\frac{\sin a+2\sin a\cos a}{1+\cos a+\cos 2a}$$

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