What is factor e^{jθ}-e^{-jθ} ?

The solution to factor e^{jθ}-e^{-jθ} is e^{-jθ}(e^{jθ}+1)(e^{jθ}-1)

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factor e^{jθ}-e^{-jθ}

factor

e^{j θ} - e^{- j θ}

e^{- j θ} (e^{j θ} + 1) (e^{j θ} - 1)

Solution steps

e^{j θ} - e^{- j θ}

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

e^{j θ} = e^{- j θ} e^{2 j θ}

= e^{- j θ} e^{2 j θ} - e^{- j θ}

Factor out common term

e^{- j θ}

= e^{- j θ} (e^{2 j θ} - 1)

Factor

e^{2 j θ} - 1 : (e^{j θ} + 1) (e^{j θ} - 1)

= e^{- j θ} (e^{j θ} + 1) (e^{j θ} - 1)

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What is factor e^{jθ}-e^{-jθ} ?
The solution to factor e^{jθ}-e^{-jθ} is e^{-jθ}(e^{jθ}+1)(e^{jθ}-1)