Prove that `Delta=|{:(,beta_(gamma),beta_(gamma)'+beta'_(gamma),beta'_(gamma)'),(,gamma alpha,gamma alpha'+gamma' alpha,gamma' alpha'),(,alpha beta,alpha beta'+alpha' beta,alpha' beta'):}|` `=(alphabetagamma)(alpha'beta-alpha beta')(alpha' gamma-alpha gamma')(beta' gamma-betagamma')`

The value of |{:(betagamma,betagamma'+beta'gamma,beta'gamma'),(gammaalpha,gammaalpha'+gamma'alpha,gamma'alpha'),(alphabeta,alphabeta'+alpha'beta,alpha'beta'):}| is

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)

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 that |{:(2,alpha+beta+gamma+delta,alphabeta +gammadelta),(alpha+beta+gamma+delta,2(alpha+beta)(gamma+delta),alphabeta(gamma+delta)+gammadelta(alpha+beta)),(alphabeta+gammadelta,alphabeta(gamma+delta)+gammadelta(alpha+beta),2alphabetagamma delta):}| =0

If alpha+beta=90^(@) and beta+gamma=alpha then tan alpha-tan beta=?

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

If alpha,beta,gamma,delta are the roots of x^(4)+p_(1)x^(3)+p_(2)x^(2)+p_(3)x+p_(4)=0 . Then sum of the products of roots taken two at a time " "A) qquad alpha beta+alpha gamma+alpha delta+beta gamma+beta delta+gamma delta=p_(1) " "B) alpha beta+alpha gamma+alpha delta+beta gamma+beta delta+gamma delta,=p_(2) " "C) alpha beta+alpha gamma+alpha delta+beta gamma+beta delta+gamma delta,=p_(3) " "D) alpha beta+alpha gamma+alpha delta+beta gamma+beta delta+gamma delta=p_(4)

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