Product P = sin (pi/8).cos ((5pi)/12)P=sin(π8).cos(5π12).
a. Find sin (pi/8)sin(π8) by trig identity: cos 2x = 1 - 2sin^2 xcos2x=1−2sin2x
cos (pi/4) = sqrt2/2 = 1 - 2sin^2 (pi/8)cos(π4)=√22=1−2sin2(π8)
sin^2 (pi/8) = (2 - sqrt2)/4sin2(π8)=2−√24
sin (pi/8) = sqrt(2 - sqrt2)/2sin(π8)=√2−√22 --> (sin pi/8π8 is positive)
b. Find cos ((5pi)/12) = cos tcos(5π12)=cost by identity: cos 2t = 2cos^2 t - 1cos2t=2cos2t−1
cos 2t = cos ((10pi)/12) = cos ((5pi)/6) = -sqrt3/2cos2t=cos(10π12)=cos(5π6)=−√32
sqrt3/2 = 2cos^2 t - 1√32=2cos2t−1
2cos^2 t = 1 + sqrt3/2 = (2 + sqrt3)/22cos2t=1+√32=2+√32
cos^2 t = (2 + sqrt3)/4cos2t=2+√34
cos t = cos ((5pi)/12) = sqrt(2 + sqrt3)/2cost=cos(5π12)=√2+√32--> (cos t is positive)
Finally: P = [(sqrt(2 - sqrt2}).(sqrt(2 + sqrt3))]/4P=(√2−√2).(√2+√3)4