用复数的三角形式求三角代数式的值

求$1+\dfrac{1}{1+\tan\dfrac{\pi}{2n+1}}+\dfrac{1}{1+\tan\dfrac{2\pi}{2n+1}}+\cdots +\dfrac{1}{1+\tan\dfrac{2n\pi}{2n+1}}$的值.


分析与解 记$\theta_k=\dfrac{k}{2n+1}\cdot 2\pi$,其中$k=0,1,2,\cdots, 2n$,则由$\left(\cos\theta_k+{\rm i}\sin\theta_k\right)^{2n+1}=1$可得$${\rm C}_{2n+1}^0\cos^{2n+1}\theta_k-{\rm C}_{2n+1}^2\cos^{2n-1}\theta_k\sin^2\theta_k+\cdots +(-1)^n{\rm C}_{2n+1}^{2n}\cos\theta_k\sin^{2n}\theta_k=1,$$令$x_k=\dfrac{1}{\cos\theta_k}$,移项后两边同除以$\cos^{2n+1}\theta_k$整理得到$$x_k^{2n+1}+(-1)^{n+1}{\rm C}_{2n+1}^{2n}\left(x_k^2-1\right)^n+\cdots +{\rm C}_{2n+1}^2\left(x_k^2-1\right)-{\rm C}_{2n+1}^0=0,$$这个$2n+1$次方程的$2n+1$个根恰为$x_k,k=0,1,\cdots,2n$,于是$$\sum_{k=0}^{2n}x_k=(-1)^n{\rm C}_{2n+1}^{2n}=(-1)^n\cdot \left(2n+1\right),$$又$$\cos\left(\dfrac{k}{2n+1}\cdot {2\pi}\right)=\cos\left(\dfrac{2n+1-k}{2n+1}\cdot 2\pi\right),$$于是$$\sum_{k=1}^nx_k=\dfrac 12\left(\sum_{k=1}^{2n}{x_k}\right)=\dfrac{(-1)^n(2n+1)-1}2=\begin{cases} n,& 2\mid n,\\ -(n+1),&2\nmid n.\end{cases} $$而原式等于\[\begin{split} 1+\sum_{k=1}^{2n}\dfrac{1}{1+\tan\dfrac{k\pi}{2n+1}}&=1+\sum_{k=1}^{2n}\dfrac{\cos\dfrac {k\pi}{2n+1}\left(\cos\dfrac {k\pi}{2n+1}-\sin\dfrac{k\pi}{2n+1}\right)}{\cos\left(\dfrac{k}{2n+1}\cdot 2\pi\right)}\\
&=1+\dfrac 12\sum_{k=1}^n{\left(1-\tan\dfrac{2k\pi}{2n+1}+\dfrac 1{\cos{\dfrac {2k\pi}{2n+1}}}\right)}\\
&=(n+1)+\sum_{k=1}^{n}x_k\\
&=\begin{cases} 2n+1,& 2 \mid n,\\ 0,&2\nmid n.\end{cases} \end{split} \]

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