Question

cos^2 4x+sin^2 2x-1=0

Answer 1

cos²4x+sin²2x-1=0
(cos(2*2x))²+sin²2x-1=0
(1-2sin²2x)²+sin²2x-1=0
1-4sin²2x+4sin⁴2x+sin²2x-1=0
4sin⁴2x-3sin²2x=0
sin²2x*(4sin²2x-3)=0

sin²2x=0 или 4sin²2x-3=0
1. sin²2x=0, sin2x=0. 2x=πn, n∈Z.   x=πn/2, n∈Z
2. 4sin²2x-3=0, sin²2x=3/4.  sin2x=+-√(3/4).  sin2x=+-√3/2
a. sin2x=-√3/2
$2x=(-1) ^{n}*arcsin(- frac{ sqrt{3} }{2} )+ pi n,$ n∈Z
$2x= (-1)^{n+1}* frac{ pi }{3} + pi n,$ n∈Z
$x= (-1)^{n+1}* frac{ pi }{6} + frac{ pi n}{2} ,$ n∈Z
b.  sin2x=√3/2
$x= (-1)^{n}* frac{ pi }{6} + frac{ pi n}{2},$ n∈z

ответ: 
$x_{1} = frac{ pi n}{2} , x_{2} = (-1)^{n+1}* frac{ pi }{6} + frac{ pi n}{2} , x_{3} = (-1)^{n}* frac{ pi }{6} + frac{ pi n}{2}$ n∈Z