1. yes, there is a shorter method
use Angle-Sum and -Difference Identities
sin(α + β) = sin(α) cos(β) + cos(α) sin(β)
in your case α=pi/12 and β=pi/4
sin(pi/12)cos(pi/4)+cos(pi/12)sin(pi/4)
=sin(pi/12 + pi/4)
=sin(π/3)
=sqrt(3)/2
2. as you can see, your answer is not correct
