Question

(2sin^2 4a-1)/(2ctg (П/4+4a)cos^2(5П/4-4a))

Answer 1

$frac{2sin^24a-1}{2ctg(frac{pi}{4}+4a)cos^2(frac{5pi}{4}-4a)}=-1$

Числитель:

$2sin^24a-1 = -cos8a$ 

Знаменатель:

$2ctg(frac{pi}{4}+4a)cos^2(frac{5pi}{4}-4a)=2frac{cos(frac{pi}{4}+4a)cos^2(pi+frac{pi}{4}-4a)}{sin(frac{pi}{4}+4a)}=$ $2cos(frac{frac{pi}{2}+8a}{2})cos(frac{frac{pi}{2}-8a}{2})frac{cos(frac{pi}{4}-4a)}{sin(frac{pi}{4}+4a)}=$$(cosfrac{pi}{2}+cos8a)frac{cosfrac{pi}{4}cos4a+sinfrac{pi}{4}sin4a}{sinfrac{pi}{4}cos4a+cosfrac{pi}{4}sin4a}=frac{cos8a(sin4a+cos4a)}{sin4a+cos4a}=cos8a$

$frac{-cos8a}{cos8a}=-1$