Question

ДОКАЖИТЕ ТОЖДЕСТВО
$cos ^{4} a+sin ^{4} a=1-0,5sin ^{2} 2a$

Answer 1

$cos^4 alpha +sin^4 alpha =(cos^2 alpha)^2 +(sin^2 alpha )^2=(frac{1+cos2 alpha }{2})^2+(frac{1-cos2 alpha }{2})^2=\\=frac{(1+cos2 alpha )^2}{4}+frac{(1-cos2 alpha)^2 }{4}=frac{1+2cos2 alpha +cos^22 alpha +1-2cos2 alpha +cos^22 alpha }{4}=\\=frac{2+2cos^22 alpha }{4}=frac{2(1+cos^22 alpha )}{4}=frac{1+(1-sin^22 alpha) }{2}=frac{2-sin^22 alpha }{2}=1-0.5sin^22 alpha .$