Упр.23.3 ГДЗ Мерзляк 10 класс Базовый уровень (Алгебра)

1) sin(2a)/sin(a); 7) (sin^2(a)ctg(a))/sin(2a);
2) sin(2a)/(cos^2(a)-sin^2(a)); 8) (sin(a/4)+cos(a/4))(sin(a/4)-cos(a/4));
3) cos(2a)+sin^2(a); 9) (sin(a)cos(a))/(1-2sin^2(a));
4) sin(50°)/(2cos(25°)); 10) cos^4(?/4-a)-sin^4(?/4-a);
5) cos(2a)/(cos(a)-sin(a)); 11) ((sin(a)+cos(a))^2-sin(2a))/(cos(2a)+2sin^2(a));
6) 1-2sin^2(a/4); 12) tg(45°+a)/(1-tg^2(45°+a)).

1. $$\frac{\sin 2a}{\sin a}=\frac{2\sin a\cos a}{\sin a}=2\cos a.$$
2. $$\frac{\sin 2a}{\cos^2 a-\sin^2 a}=\frac{\sin 2a}{\cos 2a}=\tg 2a.$$
3. $$\cos 2a+\sin^2 a=(\cos^2 a-\sin^2 a)+\sin^2 a=\cos^2 a.$$
4. $$\frac{\sin 50^\circ}{2\cos 25^\circ}=\frac{\sin(2\cdot 25^\circ)}{2\cos 25^\circ}=\frac{2\sin 25^\circ\cos 25^\circ}{2\cos 25^\circ}=\sin 25^\circ.$$
5. $$\frac{\cos 2a}{\cos a-\sin a}=\frac{\cos^2 a-\sin^2 a}{\cos a-\sin a}=\frac{(\cos a-\sin a)(\cos a+\sin a)}{\cos a-\sin a}=\cos a+\sin a.$$
6. $$1-2\sin^2\frac a4=\cos\left(2\cdot\frac a4\right)=\cos\frac a2.$$
7. $$\frac{\sin^2 a\cdot \ctg a}{\sin 2a}=\frac{\sin^2 a\cdot \frac{\cos a}{\sin a}}{2\sin a\cos a}=\frac{\sin a\cos a}{2\sin a\cos a}=\frac12.$$
8. $$ \left(\sin\frac a4+\cos\frac a4\right)\left(\sin\frac a4-\cos\frac a4\right) =\sin^2\frac a4-\cos^2\frac a4 =-\left(\cos^2\frac a4-\sin^2\frac a4\right) =-\cos\frac a2. $$
9. $$\frac{\sin a\cos a}{1-2\sin^2 a}=\frac{\frac12\sin 2a}{\cos 2a}=\frac12\tg 2a.$$
10. $$ \cos^4\left(\frac\pi4-a\right)-\sin^4\left(\frac\pi4-a\right) =\left(\cos^2\left(\frac\pi4-a\right)-\sin^2\left(\frac\pi4-a\right)\right) \left(\cos^2\left(\frac\pi4-a\right)+\sin^2\left(\frac\pi4-a\right)\right) $$
    $$ =\cos\left(2\left(\frac\pi4-a\right)\right)\cdot 1 =\cos\left(\frac\pi2-2a\right) =\sin 2a. $$
11. $$ \frac{(\sin a+\cos a)^2-\sin 2a}{\cos 2a+2\sin^2 a} =\frac{\sin^2 a+2\sin a\cos a+\cos^2 a-\sin 2a}{(\cos^2 a-\sin^2 a)+2\sin^2 a} $$
    $$ =\frac{1+\sin 2a-\sin 2a}{1}=1. $$
12. $$ \frac{\tg(45^\circ+a)}{1-\tg^2(45^\circ+a)} =\frac12\tg\bigl(2(45^\circ+a)\bigr) =\frac12\tg(90^\circ+2a) =-\frac12\ctg 2a. $$

Ответ

1. $$2\cos a$$
2. $$\tg 2a$$
3. $$\cos^2 a$$
4. $$\sin 25^\circ$$
5. $$\cos a+\sin a$$
6. $$\cos\frac a2$$
7. $$\frac12$$
8. $$-\cos\frac a2$$
9. $$\frac12\tg 2a$$
10. $$\sin 2a$$
11. $$1$$
12. $$-\frac12\ctg 2a$$