Using properties of determinants. Prove that `|(sinalpha,cosalpha,cos(alpha+delta)),(sinbeta,cosbeta,cos(beta+delta)),(singamma,cosgamma,cos(gamma+delta))|=0`

Explanation

`L.H.S. = |[sin alpha, cos alpha, cos(alpha+delta)],[sin beta, cos beta, cos(beta+delta)],[sin gamma, cos gamma, cos(gamma+delta)]|`

`= |[sin alpha, cos alpha, cosalphacosdelta - sinalphasindelta],[sin beta, cos beta, cosbetacosdelta - sinbetasindelta],[sin gamma, cos gamma, cosgammacosdelta - singammasindelta]|`

Applying `C_3 ->C_3-cosdeltaC_2+sindeltaC_1`

`= |[sin alpha, cos alpha, 0],[sin beta, cos beta, 0],[sin gamma, cos gamma, 0]|`

If in a determinant, all elements in a row or column are `0`, then value of that determinant is `0`.

Here, `C_3` is `0`.

`:. |[sin alpha, cos alpha, 0],[sin beta, cos beta, 0],[sin gamma, cos gamma, 0]| = 0 = R.H.S.`