We want to show it is always possible to solve
for and , given for . For each component sinusoid, we can write
Applying this expansion to Eq.(A.2) yields
Equating coefficients gives
where and are known. We now have two equations in two unknowns which are readily solved by (1) squaring and adding both sides to eliminate , and (2) forming a ratio of both sides of Eq.(A.4) to eliminate . The results are
which has a unique solution for any values of and .
Proof Using Complex Variables
Half-Angle Tangent Identities