Question

Решить уравнение:

$sin^{2}frac{x}{4}+sin^2frac{3x}{4}=2cos^{2}frac{x}{2}$

Answer 1

$sin^2frac{x}{4}+sin^2frac{3x}{4}=2cos^2frac{x}{2}\\zamena:; ; a=frac{x}{4}; ; Rightarrow ; ; frac{3x}{4}=3a; ; ,; ; frac{x}{2}=2a; ; ,; ; x=4a\\sin^2a+sin^23a=2cos^22a\\frac{1-cos2a}{2}+frac{1-cos6a}{2}=2cdot frac{1+cos4a}{2}\\1-cos2a+1-cos6a=2+2cos4a\\2-(cos2a+cos6a)=2+2cos4a\\-2cdot cos4acdot cos2a-2, cos4a=0\\-2, cos4acdot (cos2a+1)=0\\a); ; cos4a=0; ; to ; ; 4a=frac{pi }{2}+pi n; ,; nin Z\\x=frac{pi}{2}+pi n; ,; nin Z$

$b); ; cos2a=-1; ; to ; ; 2a=pi +2pi k; ,; kin Z\\frac{x}{2}=pi +2pi k; ,; kin Z\\x=2pi +4pi k; ,; kin Z\\Otvet:; ; x=frac{pi}{2}+pi n,; ; x=2pi +4pi k; ,; ; n,kin Z; .$