Question

Докажите
$sin ^{4} alpha + cos ^{4} alpha = frac{1 + cos ^{2} 2alpha }{2}$

Answer 1

(sin^2 a)^2+(cos^2 a)^2=((1-cos 2a)/2)^2+((1+cos 2a)/2)^2=((1-2cos 2a+cos^2 2a)/4)+((1+2cos 2a+cos^2 2a)/4)=(1-2cos 2a+cos^2 2a+1+2cos 2a+cos^2 2a)/4=(2+2cos^2 2a)/4=(1+cos^2 2a)/2