Question

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

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

Answer 1

Answer: (A)

Let $A=\left|\begin{array}{ccc}{a}^2& a& 1\\ cosnx& cos\left(n+1\right)x& cos\left(n+2\right)x\\ sinnx& sin\left(n+1\right)x& sin\left(n+2\right)x\end{array}\right|$
$=a^2[\sin (n+2)x\cos (n+1)x-\cos (n+2)x\sin (n+1)x]$
$-a[\sin (n+2)x\cos nx-\cos (n+2)x\sin nx]$
$+1[\sin (n+1)x\cos nx-\cos (n+1)x\sin nx]$
$=a^2[\sin (n+2-n-1)x]-a[\sin (n+2-n)x]+[\sin (n+1-n)x]$
$=a^{2}sinx-asin2x+sinx$
Thus the value of the determinant is independent of n