# The value of the determinant

Question:

The value of the determinant

$\left|\begin{array}{ccc}a^{2} & a & 1 \\ \cos n x & \cos (n+1) x & \cos (n+2) x \\ \sin n x & \sin (n+1) x & \sin (n+2) x\end{array}\right|$ is independent of

(a) $n$

(b) $\mathrm{a}$

(c) $x$

(d) none of these

Solution:

(a) $n$

Let $A=\mathrm{nx}, B=(\mathrm{n}+1) \mathrm{x}, C=(\mathrm{n}+2) \mathrm{x}$

$\Rightarrow C-B=\mathrm{x}, B-A=\mathrm{x}, C-A=2 \mathrm{x}$

Thus, the given determinant is

\begin{tabular}{|lll}

$\mathrm{a}^{2}$ & $\mathrm{a}$ & 1

$\begin{array}{lll}\mathrm{a}^{2} & \mathrm{a} & 1\end{array}$

$\begin{array}{lll}\cos A & \cos B & \cos C\end{array}$

$\sin A \quad \sin B \quad \sin C \mid$

$=\mathrm{a}^{2}(\cos B \sin C-\cos C \sin B)-\mathrm{a} \times(\cos A \sin C-\cos C \sin A)+1 \times(\cos A \sin B-\sin A \cos B)$

$=\mathrm{a}^{2} \sin (C-B)-\mathrm{a} \sin (C-A)+\sin (B-A)$

$=\mathrm{a}^{2} \sin \mathrm{x}-\mathrm{a} \sin 2 \mathrm{x}+\sin \mathrm{x} \quad[$ Independent of $\mathrm{n}]$