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

Prove that: (cosalpha+cosbeta)^2+(sinalpha+sinbeta)^2=4cos^2((alpha-beta)/2)

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

Let A and B denote the statements A : cos alpha + cos beta + cos gamma =0 B : sin alpha + sin beta + sin gamma = 0 If cos(beta - gamma) + cos (gamma -alpha) + cos (alpha -beta) = - (3)/(2) , then

Prove the following : sin alpha+ sin beta+ sin gamma - sin (alpha+beta+gamma)= 4sin frac (alpha+beta)(2) sin frac (beta+gamma)(2) sin frac (gamma+alpha)(2) .

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)

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

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

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

If cot alpha cot beta=2 , show that (cos (alpha+beta))/(cos (alpha-beta)) = 1/3 .

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