Упр.25.3 ГДЗ Мерзляк 10 класс Углубленный уровень (Алгебра)

1) sin(?+?)cos(2?-?)/(tg(?-?)cos(?-?));
2) sin(?-?)cos(?-?/2)-sin(?/2+?)cos(?-?);
3) sin^2(?-x)+tg^2(?-x)tg^2(3?/2+x)+sin(?/2+x)cos(x-2?);
4) (sin(?/2-x)+sin(?-x))^2+(cos(3?/2-x)+cos(2?-x))^2;
5) ctg(?/2-?)(sin(3?/2-?)+sin(?+?))/(ctg(?+?)(cos(2?+?)-sin(2?-?))).

1. $$\frac{\sin(\pi+\alpha)\cos(2\pi-\alpha)}{\tg(\pi-\alpha)\cos(\pi-\alpha)}$$
   $$=\frac{(-\sin\alpha)\cos\alpha}{(-\tg\alpha)(-\cos\alpha)}$$
   $$=-\frac{\sin\alpha}{\tg\alpha}=-\frac{\sin\alpha}{\sin\alpha/\cos\alpha}=-\cos\alpha.$$

2. $$\sin(\pi-\beta)\cos\left(\beta-\frac{\pi}{2}\right)-\sin\left(\frac{\pi}{2}+\beta\right)\cos(\pi-\beta)$$
   $$=\sin\beta\cdot\sin\beta-\cos\beta\cdot(-\cos\beta)$$
   $$=\sin^2\beta+\cos^2\beta=1.$$

3. $$\sin^2(\pi-x)+\tg^2(\pi-x)\tg^2\left(\frac{3\pi}{2}+x\right)+\sin\left(\frac{\pi}{2}+x\right)\cos(x-2\pi)$$
   $$=\sin^2x+\tg^2x\cdot\ctg^2x+\cos x\cdot\cos x$$
   $$=\sin^2x+1+\cos^2x=2.$$

4. $$\left(\sin\left(\frac{\pi}{2}-x\right)+\sin(\pi-x)\right)^2+\left(\cos\left(\frac{3\pi}{2}-x\right)+\cos(2\pi-x)\right)^2$$
   $$=(\cos x+\sin x)^2+(\cos x-\sin x)^2$$
   $$=2(\sin^2x+\cos^2x)=2.$$

5. $$\frac{\ctg\left(\frac{\pi}{2}-\alpha\right)\left(\sin\left(\frac{3\pi}{2}-\alpha\right)+\sin(\pi+\alpha)\right)}{\ctg(\pi+\alpha)\left(\cos(2\pi+\alpha)-\sin(2\pi-\alpha)\right)}$$
   $$=\frac{\tg\alpha\cdot(-\cos\alpha-\sin\alpha)}{\ctg\alpha\cdot(\cos\alpha+\sin\alpha)}$$
   $$=-\frac{\tg\alpha}{\ctg\alpha}=-\tg^2\alpha.$$

Ответ

1) $$-\cos\alpha$$; 2) $$1$$; 3) $$2$$; 4) $$2$$; 5) $$-\tg^2\alpha$$.