If \alpha, \beta, \gamma \in {R}, then the determinant ∆=|\table ( {e}^{ {i} \alpha}+ {e}^{- {i} \alpha})^{2} , ( {e}^{ {i} \alpha}- {e}^{- {i} \alpha})^{2} , 4 ; ( {e}^{ {i} \beta}+ {e}^{- {i} \beta})^{2} , ( {e}^{ {i} \beta}- {e}^{- {i} \beta})^{2} , 4 ; ( {e}^{ {i} \gamma}+ {e}^{- {i} \gamma})^{2} , ( {e}^{ {i} \gamma}- {e}^{- {i} \gamma})^{2} , 4| is

Equal to e^{i(\alpha+\beta+\gamma)}

Independent of \alpha, \beta and \gamma