Past exam of the mathematical methods I course of the University of Oxford / 2017 resit / 1 / c / Solution

Factor out the same half-angle phase from the numerator and denominator:

\frac{1 + e ^{- i θ}}{1 - e ^{- i θ}} = \frac{e ^{- i θ /2} ( e ^{i θ /2} + e ^{- i θ /2} )}{e ^{- i θ /2} ( e ^{i θ /2} - e ^{- i θ /2} )} = - i cot \frac{θ}{2} .

Here Euler's formula converts the bracketed sums to cosine and sine. Both sides require

θ \in / 2 π Z

; at the excluded angles the original denominator vanishes.

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