Sinx+sin2x+sin3x/(cosx+cos2x+cos3x)=tg2x

[latex]\frac{sinx+sin2x+sin3x}{cosx+cos2x+cos3x}=tg2x\\\\sinx+sin2x+sin3x=sin(2x-x)+sin2x+sin(2x+x)=\\sin2x+sin2xcosx-cos2xsinx+sin2xcosx+cos2xsinx=\\sin2x+2sin2xcosx=sin2x(1+2cosx);\\\\cosx+cos2x+cos3x=cos(2x-x)+cos2x+cos(2x+x)=\\cos2x+cos2xcosx+sin2xsinx+cos2xcosx-sin2xsinx=\\cos2x+2cos2xcosx=cos2x(1+2cosx);\\\\\frac{sin2x(1+cosx)}{cos2x(1+cos2x)}=\frac{sin2x}{cos2x}=tg2x[/latex]