Product #P = sin (pi/8).cos ((5pi)/12)#.
a. Find #sin (pi/8)# by trig identity: #cos 2x = 1 - 2sin^2 x#
#cos (pi/4) = sqrt2/2 = 1 - 2sin^2 (pi/8)#
#sin^2 (pi/8) = (2 - sqrt2)/4#
#sin (pi/8) = sqrt(2 - sqrt2)/2# --> (sin #pi/8# is positive)
b. Find #cos ((5pi)/12) = cos t# by identity: #cos 2t = 2cos^2 t - 1#
#cos 2t = cos ((10pi)/12) = cos ((5pi)/6) = -sqrt3/2#
#sqrt3/2 = 2cos^2 t - 1#
#2cos^2 t = 1 + sqrt3/2 = (2 + sqrt3)/2#
#cos^2 t = (2 + sqrt3)/4#
#cos t = cos ((5pi)/12) = sqrt(2 + sqrt3)/2#--> (cos t is positive)
Finally: # P = [(sqrt(2 - sqrt2}).(sqrt(2 + sqrt3))]/4#
