Question

Sin^4 x+cos^4 x=cos^2 2x+1/4

Answer 1

(1-сos2x)²/4+(1+cos2x)²/4-cos²2x-1/4=0
1-2cos2x+cos²2x+1+2cos2x+cos²2x-4cos²2x-1=0
1-2cos²2x=0
cos²2x=1/2
cos2x=-1/2⇒2x=+-2π/3+2πk⇒x=+-π/3+πk,k∈z
cos2x=1/2⇒2x=+-π/3+2πk⇒x=+-π/6+πk,k∈z

Answer 2

Ответ:

$frac{pi }{8} +frac{pi n }{4} ,~ninmathbb {Z}$

Объяснение:

$sin^{4} x+cos^{4} x= cos^{2} 2x+frac{1}{4} ;$

Воспользуемся формулами понижения степеней:

$sin^{2} x=frac{1-cos2x}{2} ;cos^{2} x=frac{1+cos2x}{2} .$

Получим

$(frac{1-cos2x}{2} )^{2} +(frac{1+cos2x}{2} ) ^{2} = cos^{2} 2x+frac{1}{4} ;\\frac{1-2cos2x+cos^{2} 2x}{4} +frac{1+2cos2x+cos^{2}2x }{4} = cos^{2} 2x+frac{1}{4}|*4 ;\\1-2cos2x+cos^{2} 2x+1+2cos2x+cos^{2} 2x= 4cos^{2} 2x+1;\\2+2cos^{2} 2x=4cos^{2} 2x+1;\\2cos^{2} 2x=1;\1+cos4x=1;\\cos4x=0;\\4x=frac{pi }{2} +pi n,~ninmathbb {Z}\\x=frac{pi }{8} +frac{pi n }{4} ,~ninmathbb {Z}$