Question

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

• A. $n$
• B. $a$
• C. $x$
• D. $a,n$ and $x$

Answer 1

The correct option is A $n$

$D=a^2\left(\cos \right(n+1\left)x\sin \right(n+2)x-\cos (n+2\left)x\sin \right(n+1\left)x\right)$

$-a\left(\cos nx\sin \right(n+2)x-\cos (n+2\left)x\sin \right(nx\left)\right)+\cos \left(nx\right)\sin (n+1)x-\cos (n+1)x\sin \left(nx\right)$

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

$\therefore$ Independent of $n$