Question

Довести тотожність
1-(sin^6a+cos^6a)=3sin^2acos^2a
Помогите пожалуйста,буду очень благодарен

Answer 1

$1-(sin ^{6} alpha +cos ^{6} alpha )=cos ^{2} alpha +sin ^{2} alpha -sin ^{6} alpha -cos ^{6} alpha =$
$=cos ^{2} alpha (1-cos ^{4} alpha )+sin ^{2} alpha (1-sin ^{4} alpha )=$
$=cos ^{2} alpha(1-cos ^{2} alpha )(1+cos ^{2} alpha )+sin ^{2} alpha (1-sin ^{2} alpha )(1+sin ^{2} alpha )=$
$=cos ^{2} alpha sin ^{2} alpha (1+cos ^{2} alpha )+sin ^{2} alpha cos ^{2} alpha (1+sin ^{2} alpha )=$
$=cos ^{2} alpha sin ^{2} alpha (1+cos ^{2} alpha +1+sin ^{2} alpha )=3sin ^{2} alpha cos ^{2} alpha.$