Question

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

• A. n
• B. a
• C. x
• D. None of these

Answer 1

Answer: (A)

Let $A=\begin{vmatrix}a^{2}&a&1\\ cos\,nx&cos \left(n+1\right)x&cos\left(n+2\right)x\\ sin\,nx&sin \left(n+1\right)x&sin \left(n+2\right)x\end{vmatrix}$
$=a^{2}[\sin (n+2) x \cos (n+1) x-\cos (n+2) x \sin (n+1) x]$
$-a[\sin (n+2) x \cos n x-\cos (n+2) x \sin n x]$
$+1[\sin (n+1) x \cos n x-\cos (n+1) x \sin n x]$
$=a^{2}[\sin (n+2-n-1) x]-a[\sin (n+2-n) x] +[\sin (n+1-n) x]$
$=a^{2}\,sin \,x-a\,sin\, 2x+sin\,x$
Thus the value of the determinant is independent of n