From "Sum and Difference Formulas"
sin (x+y)=sin x cos y + cos x sin ysin(x+y)=sinxcosy+cosxsiny
sin (x-y)=sin x cos y - cos x sin ysin(x−y)=sinxcosy−cosxsiny
Add the equations to obtain
sin (x+y)+sin(x-y)=2*sin x cos ysin(x+y)+sin(x−y)=2⋅sinxcosy
so that, after dividing both sides by 2
sin x cos y=1/2*sin(x+y)+1/2*sin(x-y)sinxcosy=12⋅sin(x+y)+12⋅sin(x−y)
at this point , Let x=pi/6x=π6 and y=(3pi)/4y=3π4
sin x cos y=1/2*sin(x+y)+1/2*sin(x-y)sinxcosy=12⋅sin(x+y)+12⋅sin(x−y)
sin (pi/6) cos ((3pi)/4)=1/2*sin(pi/6+(3pi)/4)+1/2*sin(pi/6-(3pi)/4)sin(π6)cos(3π4)=12⋅sin(π6+3π4)+12⋅sin(π6−3π4)
sin (pi/6) cos ((3pi)/4)=1/2*sin((11pi)/12)+1/2*sin((-7pi)/12)sin(π6)cos(3π4)=12⋅sin(11π12)+12⋅sin(−7π12)
but sin((-7pi)/12)=-sin((7pi)/12)sin(−7π12)=−sin(7π12)
therefore
sin (pi/6) cos ((3pi)/4)=1/2*sin((11pi)/12)-1/2*sin((7pi)/12)sin(π6)cos(3π4)=12⋅sin(11π12)−12⋅sin(7π12)
God bless America !