Given complex number
8+9i
=\sqrt145(\cos(\tan^{-1}(9/8))+i\sin(\tan^{-1}(9/8)))
-5+6i
=\sqrt61(\cos(\pi-\tan^{-1}(6/5))+i\sin(\pi-\tan^{-1}(6/5)))
=\sqrt61(-\cos(\tan^{-1}(6/5))+i\sin(\tan^{-1}(6/5)))
Now, adding both the complex numbers we get
(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)))
=\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))}
=\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))}
=\sqrt145\(8/\sqrt145)-\sqrt61(5/\sqrt61)+i{\sqrt145(9/\sqrt145)+\sqrt61(6/\sqrt61)}
=8-5+i(9+6)
=3+15i