Let us solve the determinant.

We know that,

Determinant of 3 × 3 matrix is given as,

So,

By trigonometric identity, we have

sin (α – β) = sin α cos β – cos α sin β

So, we can write

Note that, the result has ‘a’ as well as ‘x’, but doesn’t contain ‘n’.

Thus, the determinant is independent of n.