" 7.The value of "|[1,cos(beta-a)cos(y-a)],[cos(a-beta),1],[cos(a-r)cos(theta-r),1]|

Without expanding, show that the following determinants vanish: {:|(1,cos(beta-alpha),cos(gamma-alpha)),(cos(alpha-beta),1,cos(gamma-beta)),(cos(alpha-gamma)),cos(beta-gamma),1|

A=[(1, cos (beta-alpha),cos(gamma-alpha)),(cos (alpha-beta), 1, cos(gamma-beta)),(cos (alpha-gamma),cos (beta-gamma),1)]=

det [[1, cos (beta-alpha), cos (gamma-alpha) cos (alpha-beta), 1, cos (gamma-beta) cos (beta-alpha), cos (beta-gamma), 1]] =

IF alpha , beta are the roots of the equation ax ^2 + bx +c=0 , then the value of the determinant |(1, cos ( beta - alpha ), cos alpha ),( cos (alpha- beta ),1,cos beta ),( cos alpha , cos beta , 1)|

|[1,cos(alpha-beta), cos alpha] , [cos(alpha-beta),1,cos beta] , [cos alpha, cos beta, 1]|