If `x=sin(alpha-beta)*sin(gamma-delta), y=sin(beta-gamma)*sin(alpha-delta), z=sin(gamma-alpha) *sin(beta-delta)`, then :

If cos(alpha+beta)*sin(gamma+delta)=cos(alpha-beta)*sin(gamma-delta), prove that cotalphacotbetacotgamma=cotdelta

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)=6, then find 1/5 sum_(r=1)^25 f(r) ,

" 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(0)=2 theb find the value of Sigma_(r=1)^(30) | f(r )| .

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)=6, then find 1/5 sum_(r=1)^25 f(r) ,

If cos(alpha+beta)sin(gamma+delta)="cos"(alpha-beta)"sin"(gamma-delta) , prove that cotalphacotbetacotgamma=cotdelta

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

Prove that: sinalpha+sinbeta+singamma-sin(alpha+beta+gamma)=4sin((alpha+beta)/2)sin((beta+gamma)/2)sin((gamma+alpha)/2)dot

Prove that |((beta+gamma-alpha-delta)^4,(beta+gamma-alpha-delta)^2,1),((gamma+alpha-beta-delta)^4,(gamma+alpha-beta-delta)^2,1),((alpha+beta-gamma-delta)^4,(alpha+beta-gamma-delta)^2,1)|=-64(alpha-beta)(alpha-gamma)(alpha-delta)(beta-delta)(gamma-delta)

The value of the determinant Delta = |(sin 2 alpha,sin (alpha + beta),sin (alpha + gamma)),(sin (beta + gamma),sin 2 beta,sin (gamma + beta)),((sin gamma + alpha),sin (gamma + beta),sin 2 gamma)| , is