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

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

if alpha+beta+gamma=2pi prove that cos^2alpha+cos^2beta+cos^2gamma-2cosalphacosbetacosgamma=1

The expression cos^2(alpha+beta)+cos^2(alpha-beta)-cos2alpha.cos2beta, is

If alpha,beta,gamma are the angles of a triangle and system of equations cos(alpha-beta)x+cos(beta-gamma)y+cos(gamma-alpha)z=0 cos(alpha+beta)x+cos(beta+gamma)y+cos(gamma+alpha)z=0 sin(alpha+beta)x+sin(beta+gamma)y+sin(gamma+alpha)z=0 has non-trivial solutions, then triangle is necessarily a. equilateral b. isosceles c. right angled "" d. acute angled

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

Prove that cos (alpha+beta) +sin (alpha-beta)=2sin(pi/4+alpha)cos(pi/4+beta)

If alpha,beta,gamma are three real numbers and A=[(1,"cos"(alpha-beta),"cos"(alpha-gamma)),(cos(beta-alpha),1,"cos"(beta-gamma)),("cos"(gamma-alpha),"cos"(gamma-beta),1)] , then which of following is/are true? a. A is singular b. A is symmetric c. A is orthogonal d. A is not invertible

IF alpha,beta and gamma,delta be the roots of the equation x^2+px-r=0 and x^2+px+r=0 respectively, prove that (alpha-gamma)(alpha-delta)=(beta-gamma)(beta-delta)