Prove that `|((beta+gamma-alpha-delta)^4,(beta+gamma-alpha-delta)^2,1),((gamma+alpha-beta-delta)^4,(gamma+alpha-beta-delta)^2,1),((alpha+beta-gamma-delta)^4,(alpha+beta-gamma-delta)^2,1)|=-64(alpha-beta)(alpha-gamma)(alpha-delta)(beta-delta)(gamma-delta)`

If |((beta+gamma-alpha -delta)^4 , (beta+gamma-alpha-delta)^2,1),((gamma+alpha-beta-delta)^4, (gamma+alpha-beta-delta)^2,1),((alpha+beta-gamma-delta)^4, (alpha + beta-gamma-delta)^2,1)|=-k(alpha -beta)(alpha -gamma)(alpha-delta)(beta-gamma)(beta-delta)(gamma-delta) , then the value of (k)^(1//2) is ____

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

Prove that |2alpha+beta+gamma+deltaalphabeta+gammadeltaalpha+beta+gamma+delta2(alpha+beta)(gamma+delta)alphabeta(gamma+delta)+gammadelta(alpha+beta)alphabeta+gammadeltaalphabeta(gamma+delta)+gammadelta(alpha+beta)2alphabetagammadelta|=0

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

If =(beta-gamma)(alpha-delta),y=(gamma-alpha)(beta-delta),z=(alpha-beta)(gamma-delta) ,then value of x^(3)+y^(3)+z^(3)-3xyz is

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

If x=sin(alpha-beta)*sin(gamma-delta), y=sin(beta-gamma)*sin(alpha-delta), z=sin(gamma-alpha)*sin(beta-delta) , then :

If cos (alpha + beta) sin (gamma + delta) = cos (alpha - beta) sin (gamma - delta) then: