Question

Докажите тождество ................

Answer 1

Ответ:

Объяснение:

9) Числитель 1 - 2sin^2 a = cos 2a

Знаменатель

$2ctg(\frac{\pi }{4} -a)*cos^2(\frac{\pi }{4} + a)=2\frac{cos(\pi/4 -a)}{sin(\pi/4 -a)} *cos^2(\pi/4 +a)=\\ =2\frac{cos(\pi/4 )cos(a)+sin(\pi/4)sin(a)}{sin(\pi/4)cos(a)-cos(\pi/4)sin(a)} *(cos(\pi/4 )cos(a)-sin(\pi/4)sin(a))^2=$

$=2\frac{1/\sqrt{2}*cos(a)+1/\sqrt{2}*sin(a)}{1/\sqrt{2}*cos(a)-1/\sqrt{2}*sin(a)} *(1/\sqrt{2}*cos(a)-1/\sqrt{2}*sin(a))^2=\\ =2(1/\sqrt{2}*cos(a)+1/\sqrt{2}*sin(a))*(1/\sqrt{2}*cos(a)-1/\sqrt{2}*sin(a))=$

$=2*1/\sqrt{2}*1/\sqrt{2} (cos(a)+sin(a))(cos(a)-sin(a))=\\ =2*1/2*(cos^2(a)-sin^2(a))=cos 2a$

Числитель равен знаменателю, значит, вся дробь равна 1.

10) Есть формулы: $tg(a)=\frac{2tg(a/2)}{1-tg^2(a/2)} ; ctg(a)=\frac{1-tg^2(a/2)}{2tg(a/2)}$

$2tg\frac{a}{2}*(\frac{2tg(a/2)}{1-tg^2(a/2)} +\frac{1-tg^2(a/2)}{2tg(a/2)} )*(1-tg^2(a/2))=\\ =2tg\frac{a}{2}*\frac{4tg^2(a/2)+(1 - tg^2(a/2))^2}{2tg\frac{a}{2}*(1 - tg^2(a/2))}*(1 - tg^2(a/2) )=$

$=2tg(\frac{a}{2} )*\frac{4tg^2(a/2)+1-2tg^2(a/2)+tg^4(a/2)}{2tg(a/2)*(1-tg^2(a/2))} *(1-tg^2(a/2)) =\\ =4tg^2(a/2)+1-2tg^2(a/2)+tg^4(a/2)=(1+tg^2(a/2))^2 = \frac{1}{cos^4(a/2)}$