A. n
Let us solve the determinant.
\(\begin{vmatrix} a^2 & a & 1 \\[0.3em] cos\,nx & cos(n+1)x & cos(n+2)x \\[0.3em] sin\,nx &sin(n+1)x &sin(n+2)x \end{vmatrix}\)
We know that,
Determinant of 3 × 3 matrix is given as,
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.