If $e^{i x}$ is a solution of the equation $z^n+p_1 z^{n-1}+p_2 z^{n-2}+\ldots+p_n=0$,where $p_i$ are real $(i=1, 2, \ldots, n)$,then $p_n \sin nx + p_{n-1} \sin(n-1)x + \ldots + p_1 \sin x + \sin(0) = $ (Note: The constant term in the equation is $p_n$ and the coefficient of $z^0$ is $1$ if we normalize,but here the equation is given as $z^n + p_1 z^{n-1} + \ldots + p_n = 0$. Let us assume the constant term is $p_n$. The expression to evaluate is $p_n \sin nx + p_{n-1} \sin(n-1)x + \ldots + p_1 \sin x + \sin(0)$). Given the standard form,find the value of $p_n \sin nx + p_{n-1} \sin(n-1)x + \ldots + p_1 \sin x$.

• A
  $\cos(n+1)x$
• B
  $\sin(n(n+1))x$
• C
  $0$
• D
  $1$