#"Let,"z_1=-1-7i,#
#Re(z_1)=-1," "Im(z_1)=-7#
#r_1=sqrt((-1)^2+(-7)^2) = sqrt50=5sqrt2#
#theta_1=tan^-1((-7)/(-1))=pi+tan^-1(7)#
#"Let," z_2=-3-4i#
#Re(z_2)=-3," "Im(z_2)=-4#
#r_2=sqrt((-3)^2+(-4)^2) = sqrt25=5#
#theta_2=tan^-1((-4)/(-3))=pi+tan^-1(4/3)#
#z_1=r_1cistheta_1#
#z_2=r_2cistheta_2#
#z_1z_2=(r_1cistheta_1)(r_2cistheta_2)#
#z_1z_2=r_1r_2cistheta_1cistheta_2#
By De-Moivre's theorem
#cistheta_1cistheta_2=cis(theta_1+theta_2)#
Thus,
#z_1z_2=r_1r_2cis(theta_1+theta_2)#
Substituting,
#r_1r_2=5sqrt2xx5=10sqrt2#
#theta_1+theta_2=pi+tan^-1(7)+pi+tan^-1(4/3)#
#=2pi+tan^-1(7)+tan^-1(4/3)#
#tan^-1(7)+tan^-1(4/3)=tan^-1((7+4/3)/(1-7xx4/3))#
#(7+4/3)/(1-7xx4/3)=(7xx3+4)/(3-7xx4)=(21+4)/(3-28)=25/-25=1/-1#
#tan^-1(7)+tan^-1(4/3)=tan^-1(1/-1)=2pi-tan^-1(1)#
#tan^-1(1)=pi/4#
#2pi-tan^-1(1)=2pi-pi/4=(7pi)/4#
#r_1cistheta_1r_2cistheta_2=10sqrt2cis(7pi)/4#
Thus,
#(-1-7i)(-3-4i)=10sqrt2cis(7pi)/4#
