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

If cos alpha+ cos beta + cos gamma = sin alpha+ sin beta +sin gamma= 0 , then :

If cosalpha+cosbeta+cosgamma=0=sinalpha+sinbeta+singamma , then which of the following is/are true:- (a) cos(alpha-beta)+cos(beta-gamma)+cos(gamma-delta)=-3/2 (b) cos(alpha-beta)+cos(beta-gamma)+cos(gamma-delta)=-1/2 (c) sumcos2alpha+2cos(alpha+beta)+2cos(beta+gamma)+2cos(gamma+alpha)=0 (d) sumsin2alpha+2sin(alpha+beta)+2sin(beta+gamma)+2sin(gamma+alpha)=0

If cos (alpha+beta) sin (gamma+delta)=cos (alpha-beta) sin (gamma-delta) , prove that : cot alpha cot beta cot gamma= cot delta .

Using vectors, prove that cos(alpha-beta)=cosalpha cosbeta+sinalphasinbeta

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

If alpha,beta,gamma are direction - angles of a line prove that cos2 alpha+cos2beta+cos2gamma=-1

If cos alpha , cos beta, cos gamma are the direction-cosines of a line, then the value of sin^2alpha + sin^2 beta + sin^2gamma =

Prove by vectors that : cos (alpha - beta) = cos alpha cos beta+ sin alpha sin beta .

Using vectors, prove that sin (alpha+beta)=sin alpha cos beta+ cos alpha sin beta .

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