Question

Помогите решить уравнение
cos^2(x)-cos^2(2x)+cos^2(3x)-cos^2(4x)=0

Answer 1

(1+cos2x)/2-(1+cos4x)/2+(1+cos6x)/2-(1+cos8x)/2=0
1+cos2x-1-cos4x+1+cos6x-1-cos8x=0
(cos2x-cos8x)+(cos6x-cos4x)=0
2sin2xsin5x-2sinxsin5x=0
2sin5x(sin2x-sinx)=0
2sin5x*2sin(x/2)cos(3x/2)=0
sin5x=0⇒5x=πn⇒x=πn/5
sin(x/2)=0⇒x/2=πn⇒x=2πn
cos(3x/2)=0⇒3x/2=π/2+πn⇒x=π/3+2πn/3
Ответ x=πn/5;x=π/3+2πn/3