Упр.28.331 ГДЗ Мерзляк 11 класс Углубленный уровень (Алгебра)

1. $$\sin 3a-\sin a+\sin 7a-\sin 5a$$
   $$=(\sin 3a-\sin a)+(\sin 7a-\sin 5a)$$
   $$=2\cos 2a\sin a+2\cos 6a\sin a$$
   $$=2\sin a(\cos 2a+\cos 6a)$$
   $$=2\sin a\cdot 2\cos 4a\cos 2a$$
   $$=4\sin a\cos 2a\cos 4a.$$

2. $$\cos\left(\frac{3\pi}{2}+4a\right)+\sin(3\pi-8a)-\sin(4\pi-12a)$$
   $$=\sin 4a+\sin 8a+\sin 12a$$
   $$=2\sin 6a\cos 2a+2\sin 10a\cos 2a$$
   $$=2\cos 2a(\sin 6a+\sin 10a)$$
   $$=2\cos 2a\cdot 2\sin 8a\cos 2a$$
   $$=4\cos 2a\cos 4a\sin 6a.$$

3. $$\frac{\sin 6a+\sin 2a-\cos 2a}{\cos 2a-\cos 6a-\sin 2a}$$
   $$=\frac{2\sin 4a\cos 2a-\cos 2a}{-2\sin 4a\sin(-2a)-\sin 2a}$$
   $$=\frac{\cos 2a(2\sin 4a-1)}{\sin 2a(2\sin 4a-1)}$$
   $$=\ctg 2a.$$

4. $$\frac{\sin a-\sin 3a-\sin 5a+\sin 7a}{\cos a-\cos 3a+\cos 5a-\cos 7a}$$
   $$=\frac{2\sin 4a\cos 3a-2\sin 4a\cos a}{-2\sin 4a\sin(-3a)-2\sin 4a\sin a}$$
   $$=\frac{2\sin 4a(\cos 3a-\cos a)}{2\sin 4a(\sin 3a-\sin a)}$$
   $$=\frac{-2\sin 2a\sin a}{\sin a\cos 2a}$$
   $$=-\tg 2a.$$

5. $$\left(\frac{\sin a}{\sin 2a}-\frac{\cos a}{\cos 2a}\right)\cdot\frac{\cos a-\cos 7a}{\sin a}$$
   $$=\frac{\sin a\cos 2a-\cos a\sin 2a}{\sin 2a\cos 2a}\cdot\frac{-2\sin 4a\sin(-3a)}{\sin a}$$
   $$=\frac{\sin(-a)}{\sin 2a\cos 2a}\cdot\frac{2\sin 4a\sin 3a}{\sin a}$$
   $$=\frac{-\sin a\cdot 2\sin 4a\sin 3a}{\sin a\sin 2a\cos 2a}$$
   $$=\frac{-2\sin 4a\sin 3a}{\sin 2a\cos 2a}$$
   $$=-4\sin 3a.$$

6. $$\cos^2\left(\frac{5\pi}{8}+a\right)-\sin^2\left(\frac{15\pi}{8}+a\right)$$
   $$=\frac{1+\cos\left(\frac{5\pi}{4}+2a\right)}{2}-\frac{1-\cos\left(\frac{15\pi}{4}+2a\right)}{2}$$
   $$=\frac{\cos\left(\frac{5\pi}{4}+2a\right)+\cos\left(\frac{15\pi}{4}+2a\right)}{2}$$
   $$=\frac{2\cos\left(\frac{5\pi}{2}+2a\right)\cos\frac{5\pi}{4}}{2}$$
   $$=\cos\left(\frac{5\pi}{2}+2a\right)\cdot\left(-\frac{\sqrt2}{2}\right)$$
   $$=\frac{\sqrt2}{2}\sin 2a.$$

Ответ

1) $$4\sin a\cos 2a\cos 4a$$;
2) $$4\cos 2a\cos 4a\sin 6a$$;
3) $$\ctg 2a$$;
4) $$-\tg 2a$$;
5) $$-4\sin 3a$$;
6) $$\frac{\sqrt2}{2}\sin 2a$$.