Question

$\left|\begin{array}{ccc}1& 1& 1\\ cos\left(nx\right)& cos(n+1)x& cos(n+2)x\\ sin\left(nx\right)& sin(n+1)x& sin(n+2)x\end{array}\right|$ does not depend

• A. On n
• B. On x
• C. Both on x and n
• D. Either on x or on n

Answer 1

The correct option is A On n

$△=\left|\begin{array}{ccc}1& 1& 1\\ cosnx& cos(n+1)x& cos(n+2)x\\ sinnx& sin(n+1)x& sin(n+2)x\end{array}\right|$
Applying$C_1\to C_1+C_3-\left(2cosx\right)C_2△=\left|\begin{array}{ccc}2(1-cosx)& 1& 1\\ 0& cos(n+1)x& cos(n+2)x\\ 0& sin(n+1)x& sin(n+2)x\end{array}\right|△=2(1-cosx)\left[cos\right(n+1\left)xsin\right(n+2)x-cos(n+2\left)xsin\right(n+1\left)x\right]△=2(1-cosx)\left[sin\right(n+2-n-1\left)x\right]=2sinx(1-cosx)$
$i.e.,△$ is independent of n.