Question

Without expanding evaluate the determinant `|(sinalpha,cosalpha,sin(alpha+delta)),(sinbeta,cosbeta,sin(beta+delta)),(singamma,cosgamma,sin(gamma+delta))|`

Answer 1

Let the given determinant be `Delta`. Then, applying `C_(3) to C_(3)+ ("sin" delta) C_(1) -("cos" delta)C_(2)`, we get
`Delta =|["sin"alpha, "cos"alpha, 0],["sin"beta,"cos"beta,0],["sing"gamma, "cos"gamma, 0]|=0[because "each element in "C_(3) "is"0]`