Let us write the two complex numbers in polar coordinates and let them be
#z_1=r_1(cosalpha+isinalpha)# and #z_2=r_2(cosbeta+isinbeta)#
Here, if two complex numbers are #a_1+ib_1# and #a_2+ib_2# #r_1=sqrt(a_1^2+b_1^2)#, #r_2=sqrt(a_2^2+b_2^2)# and #alpha=tan^(-1)(b_1/a_1)#, #beta=tan^(-1)(b_2/a_2)#
Their multipication leads us to
#{r_1*r_2}{(cosalpha+isinalpha)*(cosbeta+isinbeta)}# or
#{r_1*r_2}{(cosalphacosbeta+i^2sinalphasinbeta)+i(cosalphasinbeta+cosbetasinalpha))# or
#{r_1*r_2}{(cosalphacosbeta-sinalphasinbeta)+i(cosalphasinbeta+cosbetasinalpha))# or
#(r_1*r_2)*(cos(alpha+beta)+isin(alpha+beta))# or
#z_1*z_2# is given by #(r_1*r_2, (alpha+beta))#
So for multiplication of complex number #z_1# and #z_2# , take new angle as #(alpha+beta)# and modulus os #r_1*r_2# of the modulus of two numbers.
Here #7-3i# can be written as #r_1(cosalpha+isinalpha)# where #r_1=sqrt(7^2+(-3)^2)=sqrt58# and #alpha=tan^(-1)((-3)/7)#
and #5-i# can be written as #r_2(cosbeta+isinbeta)# where #r_2=sqrt(5^2+(-1)^2)=sqrt26# and #beta=tan^(-1)(-1/5)#
and #z_1*z_2=sqrt58*(sqrt26)(costheta+isintheta)#, where #theta=alpha+beta#
Hence, #tantheta=tan(alpha+beta)=(tanalpha+tanbeta)/(1-tanalphatanbeta)=((-3)/7+(-1/5))/(1-((-3)/7xx(-1/5)))=(-22/35)/(32/35)=-22/32=-11/16#.
Hence, #(7-3i)(5-i)=sqrt(58xx26)(costheta+isintheta)#
= #2sqrt377(costheta+isintheta)#, where #theta=tan^(-1)(-11/16)#
