If `alpha,beta,gamma` do not differ by a multiple of `pi` and if `cos(alpha+theta)/sin(beta+gamma)=cos(beta+theta)/sin(gamma+alpha)=cos(gamma+theta)/sin(alpha+beta)=k` Then k equals to

If alpha, beta , gamma do not differ by a multiple of a pi and if ( cos (alpha + theta))/( sin (beta + gamma)) = (cos (bega + theta))/( sin (gamma + alpha )) = (cos (gamma + theta))/( sin (alpha + beta)) = k. Then k equals

cos alpha sin (beta-gamma)+cos beta sin (gamma-alpha) +cos gamma(sin alpha-beta)=

cos alpha sin (beta-gamma) + cos beta sin (gamma-alpha) + cos gamma sin (alpha-beta)=

cos alpha * sin (beta - gamma ) + cos beta * sin ( gamma - alpha ) + cos gamma * sin ( alpha - beta )=

cos alpha sin (beta-gamma) + cos beta sin (gamma-alpha) + cos gamma sin (alpha-beta) =

cos alpha sin (beta-gamma)+cos betasin (gamma-alpha+cos gamma sin (alpha-beta)=

sin(beta+gamma-alpha)+sin(gamma+alpha-beta)+sin(alpha+beta-gamma)-sin(alpha+beta+gamma)=

cos (alpha + beta + gamma ) + cos (alpha - beta - gamma) + cos ( beta - gamma - alpha ) + cos ( gamma - alpha - beta )=

cos (alpha + beta) cos gamma-cos (beta + gamma) cos alpha = sin beta sin (gamma-alpha)