Problem source

The notation $\displaystyle \prod^n_{r=1} \f (r)$ 
denotes the product $\f (1) \times \f (2) \times \f(3) \times \cdots \times
\f(n)$. 
%For example, $\displaystyle \prod_{r=1}^4 r = 24$.
%Simplify $\displaystyle \prod^n_{r=1} \frac{\g (r) }{ \g (r-1) }$. 
%You may assume that $\g (r) \neq 0$ for any integer $0 \le r \le n $.
Simplify the following products as far as possible:
\begin{questionparts}
\item $\displaystyle \prod^n_{r=1} \l \frac{r+ 1 }{ r } \r\,$;
\item $\displaystyle \prod^n_{r=2} \l \frac{r^2 -1}{r^2 } \r\,$;
\item $\displaystyle \prod^n_{r=1} \l {\cos \frac{2\pi }{ n} 
+ \sin \frac{2\pi}{ n} \cot \frac{\l 2r-1 \r \pi  }{ n} }\r\,$,
 where $n$ is even.
\end{questionparts}

Solution source

\begin{questionparts}
\item $\,$
\begin{align*}
 \prod^n_{r=1} \left ( \frac{r+ 1 }{ r } \right) &= \frac{2}{1} \cdot \frac{3}{2} \cdot \frac{4}{3} \cdots \frac{n-1}{n-2} \cdot \frac{n}{n-1} \cdot \frac{n+1}{n} \\
&= \frac{n+1}{1} = n+1
\end{align*}

\item $\,$
\begin{align*}
\prod^n_{r=2} \left ( \frac{r^2 -1}{r^2 }  \right) &= \prod^n_{r=2} \left ( \frac{(r -1)(r+1)}{r^2 }  \right) \\
&= \left ( \frac{1}{2} \cdot \frac{3}{2} \right) \cdot \left ( \frac{2}{3} \cdot \frac{4}{3} \right) \cdots \left ( \frac{r-1}{r} \cdot \frac{r+1}{r}\right) \cdots \frac{n-1}{n} \cdot \frac{n+1}{n} \\
&= \frac{1}{n} \cdot \frac{n+1}{2} \\
&= \frac{n+1}{2n} 
\end{align*}

\item When $n$ is odd, the product is undefined, since we have a $\cot \pi$ lurking in there.
\begin{align*}
\prod^n_{r=1} \left ( {\cos \frac{2\pi }{ n} 
+ \sin \frac{2\pi}{ n} \cot \frac{ (2r-1 ) \pi  }{ n} } \right) &= \prod^n_{r=1} \left ( {\cos \frac{2\pi }{ n} 
+ \sin \frac{2\pi}{ n} \frac{\cos \frac{ (2r-1 ) \pi  }{ n}}{\sin\frac{ (2r-1 ) \pi  }{ n}} } \right) \\
&= \prod^n_{r=1} \frac{1}{\sin\frac{ (2r-1 ) \pi  }{ n}} \left ( {\cos \frac{2\pi }{ n} \sin\frac{ (2r-1 ) \pi  }{ n}
+ \sin \frac{2\pi}{ n} \cos \frac{ (2r-1 ) \pi  }{ n} } \right) \\
&= \prod^n_{r=1} \frac{1}{\sin\frac{ (2r-1 ) \pi  }{ n}} \sin \left ( \frac{2\pi}{n} + \frac{(2r-1)\pi}{n} \right) \\
&= \prod^n_{r=1} \frac{1}{\sin\frac{ (2r-1 ) \pi  }{ n}} \sin \left ( \frac{(2r+1)\pi}{n} \right) \\
&= \frac{\sin \frac{3\pi}{n}}{\sin \frac{\pi}{n}} \cdot  \frac{\sin \frac{5\pi}{n}}{\sin \frac{3\pi}{n}} \cdots  \frac{\sin \frac{(2n+1)\pi}{n}}{\sin \frac{(2n-1)\pi}{n}} \\
&=  \frac{\sin \frac{(2n+1)\pi}{n}}{\sin \frac{\pi}{n}} \\
&= 1
\end{align*}
\end{questionparts}