Упр.23.22 ГДЗ Мерзляк 10 класс Базовый уровень (Алгебра)

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

1. $$\frac{\sin 3a}{\sin a}-\frac{\cos 3a}{\cos a}=\frac{\sin 3a\cdot \cos a-\cos 3a\cdot \sin a}{\sin a\cdot \cos a}=\frac{\sin(3a-a)}{\sin a\cos a}$$
   $$=\frac{\sin 2a}{\sin a\cos a}=\frac{2\sin a\cos a}{\sin a\cos a}=2.$$
2. $$\frac{1}{\ctg \frac a2-\tg \frac a2}=\frac{1}{\frac{\cos \frac a2}{\sin \frac a2}-\frac{\sin \frac a2}{\cos \frac a2}}=\frac{\sin \frac a2\cos \frac a2}{\cos^2 \frac a2-\sin^2 \frac a2}$$
   $$=\frac{\frac12\sin a}{\cos a}=\frac12\tg a.$$
3. $$(\tg a+\ctg a)\sin 2a=\left(\frac{\sin a}{\cos a}+\frac{\cos a}{\sin a}\right)\cdot 2\sin a\cos a$$
   $$=2(\sin^2 a+\cos^2 a)=2.$$
4. $$\frac{1}{1-\tg a}-\frac{1}{1+\tg a}=\frac{(1+\tg a)-(1-\tg a)}{(1-\tg a)(1+\tg a)}$$
   $$=\frac{2\tg a}{1-\tg^2 a}=\tg 2a.$$
5. $$\frac{4\tg a(1-\tg^2 a)}{(1+\tg^2 a)^2}=2\cdot \frac{2\tg a}{1+\tg^2 a}\cdot \frac{1-\tg^2 a}{1+\tg^2 a}$$
   $$=2\sin 2a\cdot \cos 2a=\sin 4a.$$
6. $$\frac{\tg 2a\cdot \tg a}{\tg 2a-\tg a}=\frac{\frac{2\tg a}{1-\tg^2 a}\cdot \tg a}{\frac{2\tg a}{1-\tg^2 a}-\tg a}$$
   $$=\frac{2\tg^2 a}{2\tg a-\tg a(1-\tg^2 a)}=\frac{2\tg^2 a}{\tg a(1+\tg^2 a)}=\frac{2\tg a}{1+\tg^2 a}=\sin 2a.$$
7. $$\frac{\sin^4 a-\cos^4 a+\cos^2 a}{2(1-\cos a)}=\frac{(\sin^2 a-\cos^2 a)(\sin^2 a+\cos^2 a)+\cos^2 a}{2(1-\cos a)}$$
   $$=\frac{\sin^2 a-\cos^2 a+\cos^2 a}{2(1-\cos a)}=\frac{\sin^2 a}{2(1-\cos a)}$$
   $$=\frac{1-\cos^2 a}{2(1-\cos a)}=\frac{(1-\cos a)(1+\cos a)}{2(1-\cos a)}=\frac{1+\cos a}{2}=\cos^2 \frac a2.$$
8. $$2\cos\left(\frac{\pi}{4}-\frac a2\right)\cos\left(\frac{\pi}{4}+\frac a2\right)$$
   $$=2\cos\left(\frac{\pi}{2}-\left(\frac{\pi}{4}+\frac a2\right)\right)\cos\left(\frac{\pi}{4}+\frac a2\right)$$
   $$=2\sin\left(\frac{\pi}{4}+\frac a2\right)\cos\left(\frac{\pi}{4}+\frac a2\right)=\sin\left(\frac{\pi}{2}+a\right)=\cos a.$$

Ответ

1) $$2$$; 2) $$\frac12\tg a$$; 3) $$2$$; 4) $$\tg 2a$$; 5) $$\sin 4a$$; 6) $$\sin 2a$$; 7) $$\cos^2 \frac a2$$; 8) $$\cos a$$.