Thus, equality holds when we define
Since is just the polar representation of a complex number, there is always some value of and such that equals whatever complex number results on the right-hand side of Eq.(A.5).
As is often the case, we see that the use of Euler's identity and complex analysis gives a simplified algebraic proof which replaces a proof based on trigonometric identities.
Phasor Analysis: Factoring a Complex Sinusoid into Phasor Times Carrier
Proof Using Trigonometry