Упр.20.2 ГДЗ Мерзляк 10 класс Базовый уровень (Алгебра)

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

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

Ответ

1) $$\frac{1}{\sin^2 5a}$$; 2) $$\cos \frac{a}{3}$$; 3) $$-\ctg^2 \gamma$$; 4) $$\frac{1}{\sin^2 a}$$; 5) $$1$$; 6) $$1$$; 7) $$1$$; 8) $$4$$.