The value of the determinant $\begin{vmatrix}a^2&a&1\\\cos nx &\cos (n + 1) x&\cos (n + 2) x\\\sin nx &\sin (n+1) x&\sin (n+2) x\end{vmatrix}$ is independent of

$n$

We have,

$\begin{vmatrix}a^2&a&1\\\cos nx &\cos (n + 1) x&\cos (n + 2) x\\\sin nx &\sin (n+1) x&\sin (n+2) x\end{vmatrix}$

$=a^2 \sin \{(n+2) x-(n+1) x\}-a \sin \{(n+2) x-nx\} + \sin \{(n+1)x-nx\}$

$=a^2 \sin x-a \sin 2x + \sin x$

$=(a^2-2a \cos x + 1) \sin x$

Clearly, it is independent of n.