1)4x-1=3x²
3x²-4x+1=0
D=16-12=4
x1=(4-2)/6=1/3 U x2=(4+2)/6=1
2)arcctg(2x-3)=arcctg0
2x-3=0⇒x=1,5
3)x²-x=2x-2
x²-3x+2=0
x1+x2=3 U x1*x2=2⇒x1=1 U x2=2
4)sinx=π/4⇒x=(-1)^n *arcsinπ/4 +πn
5)sinx*sin2x/cosx*cos2x=-1
cosx*cos2x≠0⇒x≠π/2+πn U x≠π/4+πn/2
sinx*sin2x+cosxcos2x=0
cosx=0-нет решения
6)sin7xsinx-sin3xsin5x=0
1/2cos6x-1/2cos8x-1/2cos2x+1/2cos8x=0
1/2cos6x-1/2cos2x=0
1/2(cos6x-cos2x)=0
1/2*(-2sin2xsin4x)=0
sin2x=0⇒2x=πn⇒x=πn/2
sin4x=0⇒4x=πn⇒x=πn/4
Ответ x=πn/4
7)sin6x +cos6xcos3x/sin3x=√3
(sin6xsin3x+cos6xcos3x)=sin3x*√3 sin3x≠0⇒x≠πn/3
cos3x-sin3x*√3=0
2(1/2cos3x-√3/2sin3x)=0
2sin(π/6-3x)=0
sin(π/6-3x)=0⇒π/6-3x=πn⇒3x=π/6+πn⇒x=π/18+πn/3
8)(sinx+cosx)²-(sinx-cosx)=0
(sinx-cosx)(sinx-cosx-1)=0
sinx-cosx=0⇒sinx-sin(π/2-x)=0⇒2sin(x-π/4)cosπ/4=0⇒sin(x-π/4)=0⇒
⇒x-π/4=πn⇒x=π/4+πn
sinx-cosx-1=0
2sinx/2cosx/2-cos²x/2+sin²x/2-cos²x/2-sin²x/2=0
2sinx/2cosx/2-2cos²x/2=0 /2cos²x/2≠0
tgx/2-1=0⇒tgx/2=1⇒x/2=π/4+πn⇒x=π/2+2πn