Question

Известно, что sin(a) + cos(a) = m. Найдите значение выражения sin^4(a) + cos^4(a)

Answer 1

$sin alpha +cos alpha =m \ \ (sin alpha +cos alpha)^2 =m^2 \ sin^2 alpha +cos^2 alpha +2sin alpha cos alpha =m^2 \ 1+sin2 alpha =m^2 \ sin2 alpha =m^2-1 \ \ sin^4 alpha +cos^4 alpha =(sin^2 alpha +cos^2 alpha )^2-2sin^2 alpha cos^2 alpha =1- frac{1}{2}sin^22 alpha = \ =1- frac{1}{2}(m^2-1)^2 =1- frac{1}{2}(m^4-2m^2+1) =1- frac{1}{2}m^4+m^2 - frac{1}{2}= \ =- frac{1}{2}m^4+m^2+ frac{1}{2}$