Докажите тождество: cos2x-cos3x-cos4x+cos5x=-4sin(x/2)cos(7x/2)sinx

Question

Докажите тождество: cos2x-cos3x-cos4x+cos5x=-4sin(x/2)cos(7x/2)sinx

Answer 1

cos(2x)-cos(3x)-cos(4x)+cos(5x)=(cos(5x)+cos(2x))-(cos(4x)+cos(3x))=

=2cos(7x/2)*cos(3x/2)-2cos(7x/2)*cos(x/2)=

=2cos(7x/2)*(cos(3x/2)-cos(x/2))=2cos(7x/2)*(-2sin(x)*sin(x/2)=

=-4sin(x/2)*cos(7x/2)*sin(x)