Упр.26.8 ГДЗ Мерзляк 10 класс Углубленный уровень (Алгебра)

1) 1-cos(5?/6); 2) 1+cos(12?); 3) 1+cos(40°);
4) 1-sin(?/2)+2cos(2?)+cos(4?)=4cos^2(?)cos(2?).

1. $$1-\cos\frac{5\alpha}{6}=1-\left(1-2\sin^2\frac{5\alpha}{12}\right)=2\sin^2\frac{5\alpha}{12}.$$

2. $$1+\cos 12\alpha=1+\left(2\cos^2 6\alpha-1\right)=2\cos^2 6\alpha.$$

3. $$1+\cos 40^\circ=1+\left(2\cos^2 20^\circ-1\right)=2\cos^2 20^\circ.$$

4. $$1-\sin\frac{\alpha}{2}+2\cos 2\alpha+\cos 4\alpha-4\cos^2\alpha\cos 2\alpha$$
   $$=1-\sin\frac{\alpha}{2}+2\cos 2\alpha+\left(2\cos^2 2\alpha-1\right)-4\cos^2\alpha\cos 2\alpha$$
   $$=-\sin\frac{\alpha}{2}+2\cos 2\alpha\left(1+\cos 2\alpha\right)-4\cos^2\alpha\cos 2\alpha$$
   $$=-\sin\frac{\alpha}{2}+2\cos 2\alpha\left(1+2\cos^2\alpha-1\right)-4\cos^2\alpha\cos 2\alpha$$
   $$=-\sin\frac{\alpha}{2}+4\cos^2\alpha\cos 2\alpha-4\cos^2\alpha\cos 2\alpha$$
   $$=-\sin\frac{\alpha}{2}.$$

Ответ

$$1)\;2\sin^2\frac{5\alpha}{12};\quad 2)\;2\cos^2 6\alpha;\quad 3)\;2\cos^2 20^\circ;\quad 4)\;-\sin\frac{\alpha}{2}.$$