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

n

The correct answer is option (1) : n

We have,

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

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

$=a^2 sin x- asin2x+sin x$

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

Clearly, it is independent of n.