Sin3x +sinx =2sin(3x+x)/2*cos(3x-x)/2 =2sin2x*cosx.
sinx -sin3x =2sin(x-3x)/2*cos(x+3x)/2 =2sin(-x)*cos2x = -2sinx*cos2x.
cos5y +cosy =2cos(5y+y)/2*cos(5y -y)/2 =2cos3y*cos2y.
cos4z+cos2z =2cos3z*cosz.
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sinx -sin3x =sinx +sin(-3x) =2sin(x+(-3x))/2*cos(x-(-3x))/2 =2sin(-x)*cos2x = -2sinx*cos2x.