Given complex number
8+9i8+9i
=\sqrt145(\cos(\tan^{-1}(9/8))+i\sin(\tan^{-1}(9/8)))=√145(cos(tan−1(98))+isin(tan−1(98)))
-5+6i−5+6i
=\sqrt61(\cos(\pi-\tan^{-1}(6/5))+i\sin(\pi-\tan^{-1}(6/5)))=√61(cos(π−tan−1(65))+isin(π−tan−1(65)))
=\sqrt61(-\cos(\tan^{-1}(6/5))+i\sin(\tan^{-1}(6/5)))=√61(−cos(tan−1(65))+isin(tan−1(65)))
Now, adding both the complex numbers we get
(8+9i)+(-5+6i)(8+9i)+(−5+6i)
=\sqrt145(\cos(\tan^{-1}(9/8))+i\sin(\tan^{-1}(9/8)))+\sqrt61(-\cos(\tan^{-1}(6/5))+i\sin(\tan^{-1}(6/5)))=√145(cos(tan−1(98))+isin(tan−1(98)))+√61(−cos(tan−1(65))+isin(tan−1(65)))
=\sqrt145\cos(\tan^{-1}(9/8))-\sqrt61\cos(\tan^{-1}(6/5))+i{\sqrt145\sin(\tan^{-1}(9/8))+\sqrt61\sin(\tan^{-1}(6/5))}=√145cos(tan−1(98))−√61cos(tan−1(65))+i{√145sin(tan−1(98))+√61sin(tan−1(65))}
=\sqrt145\cos(\cos^{-1}(8/\sqrt145))-\sqrt61\cos(\cos^{-1}(5/\sqrt61))+i{\sqrt145\sin(\sin^{-1}(9/\sqrt145))+\sqrt61\sin(\tan^{-1}(6/\sqrt61))}=√145cos(cos−1(8√145))−√61cos(cos−1(5√61))+i{√145sin(sin−1(9√145))+√61sin(tan−1(6√61))}
=\sqrt145\(8/\sqrt145)-\sqrt61(5/\sqrt61)+i{\sqrt145(9/\sqrt145)+\sqrt61(6/\sqrt61)}=√145(8√145)−√61(5√61)+i{√145(9√145)+√61(6√61)}
=8-5+i(9+6)=8−5+i(9+6)
=3+15i=3+15i