Manipulating complex exponentials

In Fourier and Z transforms it is common to encounter expressions such as $1 - e^{- j𝜃}$ . In some cases it is convenient to rewrite them as

1 - e^{- j𝜃} = \frac{e^{j𝜃 ∕ 2}}{e^{j𝜃 ∕ 2}} (1 - e^{- j𝜃}) = \frac{e^{j𝜃 ∕ 2} - e^{- j𝜃 ∕ 2}}{e^{j𝜃 ∕ 2}} = 2 j e^{- j𝜃 ∕ 2} sin (𝜃 ∕ 2) .

(A.14)

Similarly, one can write

1 + e^{- j𝜃} = \frac{e^{j𝜃 ∕ 2}}{e^{j𝜃 ∕ 2}} (1 + e^{- j𝜃}) = \frac{e^{j𝜃 ∕ 2} + e^{- j𝜃 ∕ 2}}{e^{j𝜃 ∕ 2}} = 2 e^{- j𝜃 ∕ 2} cos (𝜃 ∕ 2) .

(A.15)