Упр.26.16 ГДЗ Мерзляк 10 класс Углубленный уровень (Алгебра)

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

1. $$\frac{1}{\ctg \frac{\alpha}{2}-\tg \frac{\alpha}{2}}=\frac{\tg \frac{\alpha}{2}}{1-\tg^2 \frac{\alpha}{2}}=\frac{1}{2}\tg \alpha.$$
2. $$ (\tg \alpha+\ctg \alpha)\sin 2\alpha =\sin 2\alpha\left(\frac{\sin \alpha}{\cos \alpha}+\frac{\cos \alpha}{\sin \alpha}\right) $$
   $$ =\sin 2\alpha\cdot \frac{\sin^2 \alpha+\cos^2 \alpha}{\sin \alpha\cos \alpha} =\sin 2\alpha\cdot \frac{1}{\sin \alpha\cos \alpha} =\sin 2\alpha\cdot \frac{1}{0{,}5\sin 2\alpha}=2. $$
3. $$ \frac{4\tg \alpha\,(1-\tg^2 \alpha)}{(1+\tg^2 \alpha)^2} =4\tg \alpha\cdot \cos^4 \alpha\,(1-\tg^2 \alpha) $$
   $$ =4\sin \alpha\cos^3 \alpha-4\sin^3 \alpha\cos \alpha =4\sin \alpha\cos \alpha(\cos^2 \alpha-\sin^2 \alpha) $$
   $$ =2\sin 2\alpha\cdot \cos 2\alpha=\sin 4\alpha. $$
4. $$ \frac{\tg 2\alpha\cdot \tg \alpha}{\tg 2\alpha-\tg \alpha} =\frac{\frac{2\tg \alpha}{1-\tg^2 \alpha}\cdot \tg \alpha}{\frac{2\tg \alpha}{1-\tg^2 \alpha}-\tg \alpha} =\frac{2\tg \alpha}{2-1+\tg^2 \alpha} $$
   $$ =\frac{2\tg \alpha}{1+\tg^2 \alpha} =\frac{2\sin \alpha}{\cos \alpha}\cdot \cos^2 \alpha =2\sin \alpha\cos \alpha =\sin 2\alpha. $$
5. $$ \frac{\sin^4 \alpha-\cos^4 \alpha+\cos^2 \alpha}{2(1-\cos \alpha)} =\frac{(\sin^2 \alpha-\cos^2 \alpha)(\sin^2 \alpha+\cos^2 \alpha)+\cos^2 \alpha}{2(1-\cos \alpha)} $$
   $$ =\frac{\sin^2 \alpha-\cos^2 \alpha+\cos^2 \alpha}{2(1-\cos \alpha)} =\frac{\sin^2 \alpha}{2(1-\cos \alpha)} =\frac{1-\cos^2 \alpha}{2(1-\cos \alpha)} $$
   $$ =\frac{(1-\cos \alpha)(1+\cos \alpha)}{2(1-\cos \alpha)} =\frac{1+\cos \alpha}{2} =\cos^2 \frac{\alpha}{2}. $$
6. $$ 2\cos\left(\frac{\pi}{4}-\frac{\alpha}{2}\right)\cos\left(\frac{\pi}{4}+\frac{\alpha}{2}\right) =\cos\left(\frac{\pi}{2}\right)+\cos(-\alpha) $$
   $$ =\sin\left(\frac{\pi}{2}+\alpha\right)=\cos \alpha. $$

Ответ

1) $$\frac{1}{2}\tg \alpha$$; 2) $$2$$; 3) $$\sin 4\alpha$$; 4) $$\sin 2\alpha$$; 5) $$\cos^2 \frac{\alpha}{2}$$; 6) $$\cos \alpha$$.