Решите уравнения:
1) sinx+sin2x-cosx=2cos^2x
2)2sin2x-sin^2x=3cos^2x
3)sin4x-cos^4x=-sin^4x
4)1-sin2x=cosx/|cosx|

1.
$sin x+sin2x-cos x=2cos^2x sin x+2sin xcos x-cos x-2cos^2x=0 sin x(1+2cos x)-cos x(1+2cos x)=0 (1+2cos x)(sin x-cos x)=0 left[begin{array}$ 1+2cos x=0 sin x-cos x=0 end{array}right. left[begin{array}$ 2cos x=-1 mathrm{tg}x-1=0 end{array}right. left[begin{array}$ cos x=- frac{1}{2} mathrm{tg}x=1 end{array}right. left[begin{array}$ x_1=pm frac{2 pi }{3}+2 pi k, kin Z x_2= frac{ pi }{4}+ pi n, nin Z end{array}right.$

2.
$2sin2x-sin^2x=3cos^2x sin^2x-2sin2x+3cos^2x=0 sin^2x-4sin xcos x+3cos^2x=0 mathrm{tg}^2x-4mathrm{tg}x+3=0 (mathrm{tg}x-1)(mathrm{tg}x-3)=0 left[begin{array}$ mathrm{tg}x-1=0 mathrm{tg}x-3=0 end{array}right. left[begin{array}$ mathrm{tg}x=1 mathrm{tg}x=3 end{array}right. left[begin{array}$ x_1= frac{ pi }{4}+ pi k, kin Z x_2=mathrm{arctg}3+ pi n, nin Z end{array}right.$

3.
$sin4x-cos^4x=-sin^4x sin4x+sin^4x-cos^4x=0 2sin2xcos2x+(sin^2x-cos^2x)(sin^2x+cos^2x)=0 2sin2xcos2x+(-cos2x)cdot1=0 2sin2xcos2x-cos2x=0 cos2x(2sin2x-1)=0 left[begin{array}$ cos2x=0 2sin2x-1=0 end{array}right. left[begin{array}$ cos2x=0 sin2x=frac{1}{2} end{array}right.$
$left[begin{array}$ 2x=frac{pi}{2}+pi n 2x=(-1)^kfrac{pi}{6}+pi kend{array}right. left[begin{array}$ x=frac{pi}{4}+ frac{pi n}{2}, nin Z x=(-1)^kfrac{pi}{12}+ frac{pi k}{2} , kin Z end{array}right.$

4.
$1-sin2x= frac{cos x}{|cos x|} left[begin{array}$ left{begin{array}$ 1-sin2x= frac{cos x}{cos x} cos x textgreater 0 end{array}right. left{begin{array}$ 1-sin2x= frac{cos x}{-cos x} cos x textless 0 end{array}right. end{array}right. left[begin{array}$ left{begin{array}$ 1-sin2x=1 cos x textgreater 0 end{array}right. left{begin{array}$ 1-sin2x=-1 cos x textless 0 end{array}right. end{array}right.$
$left[begin{array}$ left{begin{array}$ -sin2x=0 cos x textgreater 0 end{array}right. left{begin{array}$ -sin2x=-2 cos x textless 0 end{array}right. end{array}right. left[begin{array}$ left{begin{array}$ sin2x=0 cos x textgreater 0 end{array}right. -sin2x neq -2 right. end{array}right. left{begin{array}$ sin2x=0 cos x textgreater 0 end{array}right.$
$left{begin{array}$ 2x=pi n cos x textgreater 0 end{array}right. left{begin{array}$ x= frac{pi n}{2} cos x textgreater 0 end{array}right. x=2 pi n, nin Z$