Question

упростить 2cos (x+pi/4)cos (2x+pi/4)+sin3x

Answer 1

$2cos(x+frac{pi}{4})cos(2x+frac{pi}{4})+sin(3x)=\ 2(cosxcosfrac{pi}{4}-sin xsin frac{pi}{4})(cos(2x)cosfrac{pi}{4}-sin(2x)sinfrac{pi}{4})+sin(3x)=\ 2(frac{sqrt{2}}{2}cosx-frac{sqrt{2}}{2}sin x)(frac{sqrt{2}}{2}cos(2x)-frac{sqrt{2}}{2}sin(2x))+sin(3x)=\ 2*frac{sqrt{2}}{2}*frac{sqrt{2}}{2}(cosx-sin x)(cos(2x)-sin(2x))+sin(3x)=\ cosxcos(2x)+sinxsin(2x)-(cos(2x)sin x+cosxsin (2x))+sin (3x)=\ cos(x-2x)-sin(2x+x)+sin(3x)=\ cos(-x)-sin (3x)+sin(3x)=cos x$