Given:
#((1-isqrt3)(cos(varphi)+isin(varphi)))/(2(1-i)(cos(varphi)-isin(varphi)))#
Using Euler's formula #e^(ivarphi)=cos(varphi)+isin(varphi)# and #e^(-ivarphi)=cos(varphi)-isin(varphi)#
#((1-isqrt3)(cos(varphi)+isin(varphi)))/(2(1-i)(cos(varphi)-isin(varphi)))= ((1-isqrt3)(e^(i(varphi))))/(2(1-i)(e^(-i(varphi))))#
Division of exponential function is the same as subtracting the exponents:
#((1-isqrt3)(e^(i(varphi))))/(2(1-i)(e^(-i(varphi)))) = ((1-isqrt3)(e^(i(varphi- -varphi))))/(2(1-i))= #
#((1-isqrt3)(e^(i(2varphi))))/(2(1-i))#
Convert #(1-isqrt3)# to #Ae^(itheta)# form:
#A = sqrt(1^2+(-sqrt3)^2) = 2# and #theta = tan^-1(-sqrt3/1)= -pi/3#
#((1-isqrt3)(e^(i(2varphi))))/(2(1-i)) = (2e^(-ipi/3)(e^(i(2varphi))))/(2(1-i))#
#2/2 to 1#:
#(2e^(-ipi/3)(e^(i(2varphi))))/(2(1-i)) = (e^(-ipi/3)(e^(i(2varphi))))/((1-i))#
Convert #(1-i)# to #Ae^(itheta)# form:
#A = sqrt(1^2+(-1)^2)= sqrt2# and #theta = tan^-1(-1/1) = -pi/4#
#(e^(-ipi/3)(e^(i(2varphi))))/((1-i))= (e^(-ipi/3)(e^(i(2varphi))))/(sqrt2e^(i(-pi/4)))#
Combine the exponents:
#(e^(-ipi/3)(e^(i(2varphi))))/(sqrt2e^(i(-pi/4)))=(e^(i(2varphi-pi/12)))/sqrt2#
Rationalize the denominator:
#(e^(i(2varphi-pi/12)))/sqrt2 = sqrt2/2e^(i(2varphi-pi/12))#
Use Euler's formula:
#sqrt2/2e^(i(2varphi-pi/12)) = sqrt2/2(cos(2varphi-pi/12)+isin(2varphi-pi/12))#
