We can start by using the double angle formula
#sin(2x) = 2sinxcosx#
#cosxsinx= 1/2sin(2x)#
∴ #cos((13π)/24)sin((13π)/24)= 1/2sin((13π)/12)#
We can use the sine half-angle identity
#sin(x/2) = ±sqrt((1 − cos x)/2)#
#sin((13π)/12) = sin(π + π/12) =-sin(π/12)#, and
#π/12 = (π/6)/2#
#π/6# is in the first quadrant, so we use the positive sign for the half-angle identity
#sin((13π)/12) = -sin((π/6)/2) = -(+sqrt((1–cos( π/6))/2))#
#sin((13π)/12)= -sqrt((1-(sqrt3)/2)/2)= -sqrt((2-sqrt3)/4)#
#sin((13π)/12) =-1/2sqrt(2-sqrt3)#
#cos((13π)/24)sin((13π)/24)= 1/2sin((13π)/12) = 1/2(-1/2sqrt(2-sqrt3))#
#cos((13π)/24)sin((13π)/24) =-1/4sqrt(2-sqrt3)#
