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

Prove that |[[alpha,beta,gamma],[alpha^2,beta^2,gamma^2],[alpha^3,beta^3,gamma^3]]|= alpha beta gamma (alpha-beta)(beta-gamma)(gamma-alpha)

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),,2alphabetagammadelta):}|=0

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)

If alpha, beta, gamma are the roots of the cubic x^(3)-px^(2)+qx-r=0 Find the equations whose roots are (i) beta gamma +1/(alpha), gamma alpha+1/(beta), alpha beta+1/(gamma) (ii) (beta+gamma-alpha),(gamma+alpha-beta),(alpha+beta-gamma) Also find the valueof (beta+gamma-alpha)(gamma+alpha-beta)(alpha+beta-gamma)

Prove that cosalpha+cosbeta+cosgamma+cos(alpha+beta+gamma)=4cos(alpha+beta)/(2)cos(beta+gamma)/2cos(gamma+alpha)/2

If alpha+beta=pi/2a n dbeta+gamma=alpha, then tanalpha equals

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

If cos^-1 alpha + cos^-1 beta + cos^-1 gamma = 3 pi , then alpha(beta + gamma) + beta (gamma + alpha) + gamma (alpha + beta) equals

If 2 sin alphacos beta sin gamma=sinbeta sin(alpha+gamma),then tan alpha,tan beta and tan gamma are in

In the fig. Prove that x=alpha+beta+gamma