We need
#cos(a+b)=cosacosb-sinasinb#
#sin(a+b)=sinacosb+sinbcosa#
#sin(pi/4)=cos(pi/4)=sqrt2/2#
#cos(pi/6)=sqrt3/2#
#sin(pi/2)=1/2#
We apply Euler's Formula
#e^(ix)=cosx+isinx#
#e^(5/4pii)=cos(5/4pi)+isin(5/4pi)#
#=-cos(1/4pi)-isin(1/4pi)#
#=-sqrt2/2-isqrt2/2#
#e^(17/12pii)=cos(17/12pi)+isin(17/12pi)#
#=cos(15/12pi+2/12pi)+isin(15/12pi+2/12pi)#
#=cos(5/4pi)cos(1/6pi)-sin(5/4pi)sin(1/6pi)+i(sin(5/4pi)cos(1/6pi)+cos(5/4pi)sin(1/6pi))#
#=-sqrt2/2*sqrt3/2+sqrt2/2*1/2+i(-sqrt2/2*sqrt3/2-sqrt2/2*1/2)#
#=(sqrt2-sqrt6)/4+i(-sqrt6-sqrt2/4)#
Therefore,
#e^(5/4pii)-e^(17/12pii)=-sqrt2/2-isqrt2/2-((sqrt2-sqrt6)/4+i(-sqrt6-sqrt2)/4)#
#=(sqrt6-3sqrt2)/4-i(sqrt6-sqrt2)/4#