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$\left| {\,\begin{array}{*{20}{c}}1&1&1\\{\cos (nx)}&{\cos (n + 1)x}&{\cos (n + 2)x}\\{\sin (nx)}&{\sin (n + 1)x}&{\sin (n + 2)x}\end{array}\,} \right|$ એ . . . પર આધારિત નથી .
$x$
$n$
બંને $x$ અને $n$
એકપણ નહી.
Solution
(b) $\Delta = \left| {\,\begin{array}{*{20}{c}}1&1&1\\{\cos nx}&{\cos (n + 1)x}&{\cos (n + 2)x}\\{\sin nx}&{\sin (n + 1)x}&{\sin (n + 2)x}\end{array}\,} \right|$
Applying ${C_1} \to {C_1} + {C_3} – (2\cos x){C_2}$
$\Delta = \left| {\,\begin{array}{*{20}{c}}{2(1 – \cos x)}&1&1\\0&{\cos (n + 1)x}&{\cos (n + 2)x}\\0&{\sin (n + 1)x}&{\sin (n + 2)x}\end{array}\,} \right|$
$\Delta = 2(1 – \cos x)[\cos (n + 1)x\sin (n + 2)x$
$ – \cos (n + 2)x\sin (n + 1)x]$
$\Delta = 2(1 – \cos x)\,[\sin (n + 2 – n – 1)x]$ $ = 2\sin x(1 – \cos x)$
i.e., $\Delta $ is independent of $n$.
