If `cos beta cos (gamma- alpha)= cos (alpha- beta+ gamma)`, prove that `cot alpha+ cot y= 2cot beta`

If cot^(2) theta= cot (theta-alpha) cot( theta- beta) , prove that cot alpha+ cot beta=2 cot 2 theta

If cos (beta - gamma) + cos (gamma - alpha) + cos (alpha - beta) = -3/2, "show that", cos alpha + cos beta + cos gamma = 0 "and" sin alpha + sin beta + singamma = 0."Hence deduce that," cos (beta - gamma) = cos (gamma - alpha) = cos (alpha - beta) = -1/2.

If cos alpha+ cos beta+ cos gamma + cos alpha cos beta cos gamma=0 , show that sin alpha (1+ cos beta cos gamma)=+- sin beta sin gamma .

Simplify: cot (beta - gamma) cot (gamma - alpha) + cot (gamma - alpha) cot (alpha - beta) + cot (alpha - beta) cot (beta - gamma)

If 2 sin alpha cos beta sin gamma =sin beta sin (alpha +gamma) then prove that tan alpha,tan beta,tan gamma are in H.P.

cos(beta-gamma)+cos(gamma-alpha)+cos(alpha-beta)=-3/2 show that cosalpha+cosbeta+cosgamma=0 and sinalpha+sinbeta+singamma=0 also cos(beta-gamma)=cos(gamma-alpha)=cos(alpha-beta)=-1/2 .

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 sin (alpha + beta) = n sin (alpha - beta) " and " n ne -1," prove that " cot alpha = (n-1)/(n+1) cot beta.

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.

If |{:(1,cos alpha, cos beta),(cos alpha, 1 , cos gamma ),(cos beta, cos gamma , 1):}|=|{:(0,cos alpha, cos beta),(cos alpha , 0 , cos gamma),(cos beta, cos gamma, 0):}| then the value of cos^2 alpha + cos^2 beta + cos^2 gamma is : (a) 1 (b) 1/2 (c) 3/8 (d) 9/4