[1]" "sin(pi/12)*cos((5pi)/8)
Think of pi/12 as (pi/6)/2.
[2]" "=sin((pi/6)/2)*cos((5pi)/8)
Half angle identity: sin(x/2)=+-sqrt((1-cos(x))/2)
**pi/12 is in the first quadrant, and sine is positive in the first quadrant. Therefore, we know that sin((pi/6)/2)=+sqrt((1-cos(pi/6))/2)
[3]" "=sqrt((1-cos(pi/6))/2)*cos((5pi)/8)
Evaluate cos(pi/6)
[4]" "=sqrt((1-sqrt3/2)/2)*cos((5pi)/8)
Think of (5pi)/8 as ((5pi)/4)/2.
[5]" "=sqrt((1-sqrt3/2)/2)*cos(((5pi)/4)/2)
Half angle identity: cos(x/2)=+-sqrt((1+cos(x))/2)
**(5pi)/8 is in the second quadrant, and cosine is negative in the second quadrant. Therefore, we know that cos(((5pi)/4)/2)=-sqrt((1+cos((5pi)/4))/2)
[5]" "=sqrt((1-sqrt3/2)/2)*(-sqrt((1+cos((5pi)/4))/2))
Evaluate cos((5pi)/4)
[6]" "=sqrt((1-sqrt3/2)/2)*(-sqrt((1+(-sqrt2/2))/2))
Simplify.
[7]" "=color(blue)(-(sqrt((1-sqrt3/2)(1-sqrt2/2)))/2)
I think you can leave it at that.
