(1)

\begin{align*} \prod_{k=1}^{n-1}\sin\frac{k\pi}{n} & =\prod_{k=1}^{n-1}\frac{e^{i\frac{k\pi}{n}}-e^{-i\frac{k\pi}{n}}}{2i}\\ & =\prod_{k=1}^{n-1}\left\{ \frac{i}{2}e^{-i\frac{k\pi}{n}}\left(1-e^{i\frac{2k\pi}{n}}\right)\right\} \\ & =\prod_{k=1}^{n-1}\left\{ \frac{i}{2}e^{-i\frac{k\pi}{n}}\lim_{x\rightarrow1}\left(x-e^{i\frac{2k\pi}{n}}\right)\right\} \\ & =\frac{1}{2^{n-1}}\prod_{k=1}^{n-1}\left\{ e^{i\frac{\pi}{2}}e^{-i\frac{k\pi}{n}}\right\} \lim_{x\rightarrow1}\frac{x^{n}-1}{x-1}\\ & =\frac{n}{2^{n-1}}e^{i\frac{\pi}{2}(n-1)}e^{-i\frac{\pi}{n}\frac{n}{2}(n-1)}\\ & =\frac{n}{2^{n-1}} \end{align*}

(1)-2

\begin{align*} \prod_{k=1}^{n-1}\sin\frac{k\pi}{n} & =\prod_{k=1}^{n-1}\frac{\pi}{\Gamma\left(\frac{k}{n}\right)\Gamma\left(1-\frac{k}{n}\right)}\\ & =\prod_{k=1}^{n-1}\frac{\pi}{\Gamma^{2}\left(\frac{k}{n}\right)}\\ & =\pi^{n-1}\lim_{z\rightarrow0}\left\{ \prod_{k=1}^{n-1}\frac{1}{\Gamma^{2}\left(z+\frac{k}{n}\right)}\right\} \\ & =\pi^{n-1}\lim_{z\rightarrow0}\left\{ \Gamma^{2}(z)\prod_{k=0}^{n-1}\frac{1}{\Gamma^{2}\left(z+\frac{k}{n}\right)}\right\} \\ & =\pi^{n-1}\lim_{z\rightarrow0}\left\{ \Gamma^{2}(z)\left(\frac{n^{nz-\frac{1}{2}}}{(2\pi)^{\frac{n-1}{2}}\Gamma(nz)}\right)^{2}\right\} \\ & =\frac{1}{n2^{n-1}}\lim_{z\rightarrow0}\left(\frac{\Gamma(z)}{\Gamma(nz)}\right)^{2}\\ & =\frac{1}{n2^{n-1}}\lim_{z\rightarrow0}\left(\frac{n\Gamma(z+1)}{\Gamma(nz+1)}\right)^{2}\\ & =\frac{n}{2^{n-1}} \end{align*}

(2)

\begin{align*} \prod_{k=1}^{n-1}\cos\frac{k\pi}{n} & =\prod_{k=1}^{n-1}\frac{e^{i\frac{k\pi}{n}}+e^{-i\frac{k\pi}{n}}}{2}\\ & =\prod_{k=1}^{n-1}\left\{ \frac{1}{2}e^{-i\frac{k\pi}{n}}\left(1+e^{i\frac{2k\pi}{n}}\right)\right\} \\ & =\prod_{k=1}^{n-1}\left\{ -\frac{1}{2}e^{-i\frac{k\pi}{n}}\lim_{x\rightarrow-1}\left(x-e^{i\frac{2k\pi}{n}}\right)\right\} \\ & =\frac{1}{2^{n-1}}\prod_{k=1}^{n-1}\left\{ -e^{-i\frac{k\pi}{n}}\right\} \lim_{x\rightarrow-1}\frac{x^{n}-1}{x-1}\\ & =\frac{1}{2^{n-1}}e^{i\pi(n-1)}e^{-i\frac{\pi}{n}\frac{n}{2}(n-1)}\frac{\left(-1\right)^{n}-1}{-2}\\ & =\frac{1}{2^{n-1}}e^{-i\frac{\pi}{2}}e^{i\frac{n\pi}{2}}\frac{e^{i\pi n}-e^{2i\pi n}}{-2}\\ & =\frac{1}{2^{n-1}}e^{i\frac{n\pi}{2}}\frac{e^{\frac{5}{2}i\pi n}-e^{\frac{3}{2}i\pi n}}{2i}\\ & =\frac{1}{2^{n-1}}\frac{e^{i\frac{n\pi}{2}}-e^{-i\frac{n\pi}{2}}}{2i}\\ & =\frac{\sin\left(\frac{n\pi}{2}\right)}{2^{n-1}} \end{align*}

(3)

\begin{align*} \prod_{k=1}^{n-1}\tan\frac{k\pi}{n} & =\left(\prod_{k=1}^{n-1}\sin\frac{k\pi}{n}\right)\left(\prod_{k=1}^{n-1}\cos\frac{k\pi}{n}\right)^{-1}\\ & =\left(\frac{n}{2^{n-1}}\right)\left(\frac{\sin\left(\frac{n\pi}{2}\right)}{2^{n-1}}\right)^{-1}\\ & =\frac{n}{\sin\left(\frac{n\pi}{2}\right)} \end{align*}