If `cos(alpha+beta)sin(gamma+delta)="cos"(alpha-beta)"sin"(gamma-delta)` , prove that `cotalphacotbetacotgamma=cotdelta`

If cosalpha+cosbeta+cosgamma=0 a n d a l so sinalpha+sinbeta+singamma=0, then prove that cos2alpha+cos2beta+cos2gamma =sin2alpha+sin2beta+sin2gamma=0 sin3alpha+sin3beta+sin3gamma=3sin(alpha+beta+gamma) cos3alpha+cos3beta+cos3gamma=3cos(alpha+beta+gamma)

If cos(alpha-beta)+cos(beta-gamma)+cos(gamma-alpha)=-3/2 , prove that cosalpha+cosbeta+cosgamma=sinalpha+sinbeta+singamma=0

If alpha,beta,gamma are real numbers, then without expanding at any stage, show that |1cos(beta-alpha)"cos"(gamma-alpha)"cos"(alpha-beta)1"cos"(gamma-beta)"cos"(alpha-gamma)"cos"(beta-gamma)1|=0

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 .

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

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

If tan beta=2sin alpha sin gamma co sec(alpha+gamma) , then cot alpha,cot beta,cotgamma are in

A line makes angles alpha,beta,gamma and delta with the diagonals of a cube, prove that cos^2alpha+cos^2beta+cos^2gamma+cos^2delta=4/3

A line makes angles alpha,""""beta,""""gamma and delta with the diagonals of a cube, prove that cos^2alpha+cos^2beta+cos^2gamma+cos^2delta=4/3