Prove that `|{:(alpha,beta,gamma),(theta,phi,psi),(lambda,mu,nu):}|=|{:(beta,mu,phi),(alpha,lambda,theta),(gamma,nu,psi):}|.`

Using properties of determinants. Prove that |(alpha,alpha^2,beta+gamma),(beta,beta^2,gamma+alpha),(gamma,gamma^2,alpha+beta)|=(beta-gamma)(gamma-alpha)(alpha-beta)(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) .

If alpha,beta "and" gamma are real number without expanding at any stage prove that |{:(1,cos(beta-alpha),cos(gamma-alpha)),(cos(alpha-beta),1,cos(gamma-beta)),(cos(alpha-gamma),cos(beta-gamma),1):}| =0.

Prove that sum_(alpha+beta+gamma = 10) (10 !)/(alpha!beta!gamma!)=3^(10)dot

Prove that |{:(1,x+alpha,y+z-alpha),(1,y+beta, z+x-beta),(1,z+gamma,x+y-gamma):}|=0

Prove that: (tan(alpha-beta)+tan(beta-lambda)+tan(lambda-alpha) = tan(alpha-beta)tan(beta-lambda)tan(lambda-alpha)

Determine the values of alpha ,beta , gamma when [{:( 0, 2beta, gamma),( alpha ,beta, -gamma),( alpha ,-beta, gamma):}] is orthogonal.

If 2(cos ( alpha-beta)+cos(beta-gamma)+cos(gamma-alpha)+3=0 , prove that (d alpha)/(sin ( beta+theta)sin(gamma+theta))+(d beta)/(sin(alpha+beta)sin(beta+theta))+(d gamma)/(sin(alpha+theta)sin(beta+theta))=0 , where, 'theta' is any real angle such that alpha+theta, beta+theta, gamma+theta are not the multiple of pi .

If alpha, beta, gamma are the cube roots of 8 , then |(alpha,beta,gamma),(beta,gamma,alpha),(gamma,alpha,beta)|=

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