Question

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

Answer 1

Ответ:

$\frac{\pi }{8} +\frac{\pi n }{4} ,~n\in\mathbb {Z}$

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

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