Упр.26.1 ГДЗ Мерзляк 10 класс Углубленный уровень (Алгебра)

1) cos(2?)+sin^2(?); 5) sin^2(?)ctg(?)/sin(2?);
2) sin(50°)/(2cos(25°)); 6) sin(?)cos(?)/(1-2sin^2(?));
3) cos(2?)/(cos(?)-sin(?)); 7) cos^4(?/4-?)-sin^4(?/4-?);
4) 1-2sin^2(?/4); 8) tg(45°+?)/(1-tg^2(45°+?)).

1. $$\cos 2\alpha+\sin^2\alpha=(\cos^2\alpha-\sin^2\alpha)+\sin^2\alpha=\cos^2\alpha.$$
2. $$\frac{\sin 50^\circ}{2\cos 25^\circ}=\frac{2\sin 25^\circ\cos 25^\circ}{2\cos 25^\circ}=\sin 25^\circ.$$
3. $$\frac{\cos 2\alpha}{\cos\alpha-\sin\alpha}=\frac{\cos^2\alpha-\sin^2\alpha}{\cos\alpha-\sin\alpha}=\cos\alpha+\sin\alpha.$$
4. $$1-2\sin^2\frac{\alpha}{4}=\cos\left(2\cdot\frac{\alpha}{4}\right)=\cos\frac{\alpha}{2}.$$
5. $$\frac{\sin^2\alpha\ctg\alpha}{\sin 2\alpha}=\frac{\sin^2\alpha\cdot\frac{\cos\alpha}{\sin\alpha}}{2\sin\alpha\cos\alpha}=\frac{1}{2}.$$
6. $$\frac{\sin\alpha\cos\alpha}{1-2\sin^2\alpha}=\frac{\frac12\sin 2\alpha}{\cos 2\alpha}=\frac12\tg 2\alpha.$$
7. $$\cos^4\left(\frac{\pi}{4}-\alpha\right)-\sin^4\left(\frac{\pi}{4}-\alpha\right)$$
   $$=\left(\cos^2\left(\frac{\pi}{4}-\alpha\right)-\sin^2\left(\frac{\pi}{4}-\alpha\right)\right)\left(\cos^2\left(\frac{\pi}{4}-\alpha\right)+\sin^2\left(\frac{\pi}{4}-\alpha\right)\right)$$
   $$=\cos\left(\frac{\pi}{2}-2\alpha\right)=\sin 2\alpha.$$
8. $$\frac{\tg(45^\circ+\alpha)}{1-\tg^2(45^\circ+\alpha)}=\frac12\tg\bigl(90^\circ+2\alpha\bigr)=-\frac12\ctg 2\alpha.$$

Ответ

1) $$\cos^2\alpha$$; 2) $$\sin 25^\circ$$; 3) $$\cos\alpha+\sin\alpha$$; 4) $$\cos\frac{\alpha}{2}$$; 5) $$\frac12$$; 6) $$\frac12\tg 2\alpha$$; 7) $$\sin 2\alpha$$; 8) $$-\frac12\ctg 2\alpha$$.