Prove that `|[alpha,beta,gamma] ,[alpha^2,beta^2,gamma^2] , [beta+gamma, gamma+alpha, beta+alpha]|` = `(alpha-beta)(beta-gamma)(gamma-alpha)(alpha+beta+gamma)`

Using properties of determinants, prove the following |(alpha,beta,gamma),(alpha^2,beta^2,gamma^2),(beta+gamma,gamma+alpha,alpha+beta)|=(alpha-beta)(beta-gamma)(gamma-alpha)(alpha+beta+gamma)

Prove that: | alpha beta gamma alpha^(2)beta^(2)gamma^(2)beta+gamma gamma+alpha alpha+beta|=(alpha-beta)(beta-gamma)(gamma-alpha)(alpha+beta+gamma)

Prove the following : |{:(alpha,alpha^(2),beta+gamma),(beta,beta^(2),gamma+alpha),(gamma,gamma^(2),alpha+beta):}|=|{:(alpha,beta, gamma),(alpha^(2),beta^(2),gamma^(2)),(beta+gamma,gamma+alpha, alpha+beta):}|=(beta-gamma)(gamma-alpha)(alpha-beta)(alpha+beta+gamma)

|[1,alpha,alpha^3],[1,beta,beta^3],[1,gamma,gamma^3]|=(alpha-beta)(beta-gamma)(gamma-alpha)(alpha+ beta+gamma)

Prove that, gammaalpha ^ (2), beta ^ (2), gamma ^ (2) beta + alpha, gamma + alpha, alpha + beta] | = (beta-gamma) (gamma-alpha) (alpha-beta) ( alpha + beta + gamma)

Using peoperties of determinants in questions 11 to 15, prove that : |{:(alpha,alpha^(2),beta+gamma),(beta,beta^(2),gamma+alpha),(gamma,gamma^(2),alpha+beta):}|=(beta-gamma)(gamma-alpha)(alpha-beta+gamma)

Show that | (1,1,1), (alpha ^ 2, beta ^ 2, gamma ^ 2), (alpha ^ 3, beta ^ 3, gamma ^ 3) | = (alpha-beta) (beta-gamma) (gamma-alpha) (alphabeta + betagamma + gammaalpha) |

Show that |(1,alpha,alpha^3),(1,beta,beta^3),(1,gamma,gamma^3)|=(alpha-beta)(beta-gamma)(gamma-alpha)( alpha+beta+gamma)

Show that |[1,alpha,alpha^2],[1,beta,beta^2],[1,gamma,gamma^2]|=(alpha-beta)(beta-gamma)(gamma-alpha)

sin (beta+ gamma- alpha) + sin (gamma+ alpha - beta) + sin (alpha + beta- gamma)- sin (alpha + beta + gamma)=