Question

помогите с тригонометрией
упростить выражение, второй пример
ответ: 1
нужно решение​

Answer 1

Ответ: 1.

Пошаговое объяснение:

$2*(sin^4\alpha+sin^2\alpha *cos^2\alpha +cos^4\alpha )^2-(sin^8\alpha +cos^8\alpha )=\\\\=2*(sin^4\alpha +2*sin^2\alpha *cos^2\alpha +cos^4\alpha-sin^2\alpha *cos^2\alpha )^2-(sin^8\alpha +cos^8\alpha )=\\\\=2*((sin^2\alpha +cos^2\alpha )^2-sin+4sin^2\alpha cos^22\alpha *cos^2\alpha )^2-(sin^8\alpha +cos^8\alpha )=\\\\=2*(1-sin^2\alpha *cos^2\alpha )^2-(sin^8\alpha +cos^8\alpha )=\\\\=2*(1-2*1*sin^2\alpha*cos^2\alpha+sin^4\alpha * cos^4\alpha )-(sin^8\alpha +cos^8\alpha ) =$

$=2-4*sin^2\alpha *cos^2\alpha +2*sin^4\alpha *cos^4\alpha -sin^8\alpha -cos^8\alpha =\\\\=2-4*sin^2\alpha *cos^2\alpha -(sin^8\alpha -2*sin^4\alpha *cos^4\alpha +cos^8\alpha )=\\\\=2-4*sin^2\alpha *cos^2\alpha -(sin^4\alpha -cos^4\alpha )^2=\\\\=2-4*sin^2\alpha *cos^2\alpha -(sin^2\alpha +cos^2\alpha )^2*(sin^2\alpha -cos^2\alpha )^2=\\\\=2-4*sin^2\alpha *cos^2\alpha -1^2*(sin^4\alpha -2*sin^2\alpha *cos^2\alpha +cos^4\alpha )=$

$=2-4*sin^2\alpha *cos^2\alpha -sin^4\alpha +2*sin^2\alpha *cos^2\alpha -cos^4\alpha =\\\\=2-(sin^4\alpha +2*sin^2\alpha *cos^2\alpha +cos^4\alpha )=\\\\=2-(sin^2\alpha +cos^2\alpha )^2=2-1^2=2-1=1.$

Answer 2

Ответ:

Пошаговое объяснение:

$1). 2(sin^4\alpha +sin^2\alpha *cos^2\alpha +cos^4\alpha )^2= \\= 2(sin^4\alpha +2*sin^2\alpha *cos^2\alpha +cos^4\alpha- sin^2\alpha *cos^2\alpha)^2=\\= 2((sin^2\alpha +cos^2\alpha )^2-sin^2\alpha *cos^2\alpha )^2 =2(1-sin^2\alpha *cos^2\alpha )^2$

2).

$(sin^8\alpha +cos^8\alpha =sin^8\alpha +2sin^4\alpha *cos^4\alpha +cos^4\alpha -2sin^4\alpha *cos^4\alpha=\\=(sin^4\alpha +cos^4\alpha )^2- 2sin^4\alpha *cos^4\alpha=\\= (sin^4\alpha +cos^4\alpha+2sin^2\alpha *cos^2\alpha -2sin^2\alpha *cos^2\alpha)^2- 2sin^4\alpha *cos^4\alpha=\\=((sin^2\alpha +cos^2\alpha )^2-2sin^2\alpha *cos^2\alpha)^2 - 2sin^4\alpha *cos^4\alpha=\\=(1-2sin^2\alpha* cos^2\alpha )^2 - 2sin^4\alpha *cos^4\alpha=$

3) Теперь вычтем из результата 1) результат 2)

=>  2(1-sin²α*cos²α)²-(1-sin²α*cos²α)² +2sin^4α*cos^4α=

$2(1-sin^2\alpha *cos^2\alpha )^2 -(1-2sin^2\alpha *cos^2\alpha )^2 +2sin^4\alpha *cos^4\alpha =\\2-4sin^2\alpha *cos^2\alpha +2*sin^4\alpha *cos^4\alpha -1+4sin^2\alpha *cos^2\alpha -4sin^4\alpha *cos^4\alpha +2sin^4\alpha *cos^4\alpha =1$