$$\begin{gathered} \mathrm{If} 1, {\mathrm{\alpha }}_1, {\mathrm{\alpha }}_2......{\mathrm{\alpha }}_{\mathrm{n}-1} \mathrm{are} \mathrm{the} {\mathrm{n}}^{\mathrm{th}} \mathrm{roots} \mathrm{of} \mathrm{unity} \\ \mathrm{and} \mathrm{n} \mathrm{is} \mathrm{an} \mathrm{odd} \mathrm{natural} \mathrm{number} \mathrm{then} \\ \left(1+{\mathrm{\alpha }}_1\right)\left(1+{\mathrm{\alpha }}_2\right).......\left(1+{\mathrm{\alpha }}_{\mathrm{n}-1}\right)= \end{gathered}$$

$1, {\mathrm{\alpha }}_1, {\mathrm{\alpha }}_2, ... {\mathrm{\alpha }}_{\mathrm{n}-1}$  are ${\mathrm{n}}^{\mathrm{th}}$ roots of unity.

$\begin{array}{rcl} & \Rightarrow & {\mathrm{x}}^{\mathrm{n}}-1=(\mathrm{x}-1) (\mathrm{x}-{\mathrm{\alpha }}_1) (\mathrm{x}-{\mathrm{\alpha }}_2) .... (\mathrm{x}-{\mathrm{\alpha }}_{\mathrm{n}-1})\end{array}$

$\mathrm{substitute} \mathrm{x}=-1,$

${\left(-1\right)}^n-1=\begin{array}{rcl} & & (-1-1) (-1-{\mathrm{\alpha }}_1) (-1-{\mathrm{\alpha }}_2) .... (-1-{\mathrm{\alpha }}_{\mathrm{n}-1})\end{array}$

$-1-1=\begin{array}{rcl} & & -2(1+{\mathrm{\alpha }}_1) (1+{\mathrm{\alpha }}_2) .... (1+{\mathrm{\alpha }}_{\mathrm{n}-1})\end{array}$

$\begin{array}{rcl}(1+{\mathrm{\alpha }}_1) (1+{\mathrm{\alpha }}_2) .... (1+{\mathrm{\alpha }}_{\mathrm{n}-1})& =& 1\end{array}$