If `f(x)=|(cos(x+alpha),cos(x+beta),cos(x+gamma)),(sin(x+alpha),sin(x+beta),sin(x+gamma)),(sin(beta+gamma),sin(gamma-alpha),sin(alpha-beta))|,` then (Given `alpha != beta != gamma)`

lf, cos (x + alpha) cos (x + beta), cos (x + gamma) sin (x + alpha), sin (x + beta), sin (x + gamma) sin (beta + gamma), sin ( gamma-alpha), sin (alpha-beta) then (Given alpha! = beta! = gamma), sin (alpha-beta)

If f(x) = |(cos(x + alpha), cos(x+beta), cos(x + gamma)),(sin(x + alpha), sin(x+beta), sin(x + gamma)), (sin(beta - gamma), sin(gamma - alpha), sin(alpha - beta))| and f(2) = -2, then |sum_(r=1)^20 f(r)| equals

If f(x) = |{:(cos (x+alpha),cos(x+beta),cos(x+gamma)),(sin (x+alpha),sin(x+beta),sin(x+gamma)),(sin(beta+gamma),sin(gamma+alpha),sin(alpha+beta)):}| then f(theta)-2f(phi)+f(psi) is equal to

If f(x) = |{:(cos (x+alpha),cos(x+beta),cos(x+gamma)),(sin (x+alpha),sin(x+beta),sin(x+gamma)),(sin(beta+gamma),sin(gamma+alpha),sin(alpha+beta)):}| then f(theta)-2f(phi)+f(psi) is equal to

If f(x) = |{:(cos (x+alpha),cos(x+beta),cos(x+gamma)),(sin (x+alpha),sin(x+beta),sin(x+gamma)),(sin(beta+gamma),sin(gamma+alpha),sin(alpha+beta)):}| then f(theta)-2f(phi)+f(psi) is equal to

If f(x) = |{:(cos (x+alpha),cos(x+beta),cos(x+gamma)),(sin (x+alpha),sin(x+beta),sin(x+gamma)),(sin(beta+gamma),sin(gamma+alpha),sin(alpha+beta)):}| then f(theta)-2f(phi)+f(psi) is equal to

If f(x)=|{:(cos(x+alpha),cos(x+beta),cos(x+gamma)),(sin(x+alpha),sin(x+beta),sin(x+gamma)),(sin(beta-gamma),sin(gamma-alpha),sin(alpha-beta)):}| and f(2) =5 "the" overset(20)underset(r=1)sumf(r) is equal to

If ,cos(x+alpha)cos(x+beta),cos(x+gamma)sin(x+alpha),sin(x+beta),sin(x+gamma)sin(beta-gamma),sin(gamma-alpha),sin(alpha-beta) and f(0)=-2, then ,sum_(r=1)^(30)|f(r)| equals

If ,cos(x+alpha)cos(x+beta),cos(x+gamma)sin(x+alpha),sin(x+beta),sin(x+gamma)sin(beta-gamma),sin(gamma-alpha),sin(alpha-beta) and f(2),=6, then find (1)/(5)sum_(r=1)^(25)f(r)

prove that | (sin alpha, sin beta, sin gamma), (cos alpha, cos beta, cos gamma), (sin 2alpha, sin 2beta, sin 2gamma) | = 4sin (1/2) (beta-gamma) * sin (1/2) (gamma-alpha) * sin (1/2) (alpha-beta) xx {sin (beta + gamma) + sin (gamma + alpha) + sin (alpha + beta)}.