# 4 Sine Pi over 10 by Cosine Pi over 5/Proof 1

## Theorem

$4 \sin \dfrac \pi {10} \cos \dfrac \pi 5 = 1$

## Proof

 $\displaystyle \paren {z + 1} \paren {z^2 - 2 z \cos \dfrac \pi 5 + 1} \paren {z^2 - 2 z \cos \dfrac {3 \pi} 5 + 1}$ $=$ $\displaystyle z^5 + 1$ Complex Algebra Examples: $z^5 + 1$ $\displaystyle \leadsto \ \$ $\displaystyle \paren {1 + i} \paren {i^2 - 2 i \cos \dfrac \pi 5 + 1} \paren {i^2 - 2 i \cos \dfrac {3 \pi} 5 + 1}$ $=$ $\displaystyle i^5 + 1$ putting $z \gets i$ $\displaystyle \leadsto \ \$ $\displaystyle \paren {1 + i} \paren {-1 - 2 i \cos \dfrac \pi 5 + 1} \paren {-1 - 2 i \cos \dfrac {3 \pi} 5 + 1}$ $=$ $\displaystyle i + 1$ Definition of Imaginary Unit $\displaystyle -4 \paren {1 + i} \cos \dfrac \pi 5 \cos \dfrac {3 \pi} 5$ $=$ $\displaystyle i + 1$ simplifying $\displaystyle -4 \cos \dfrac \pi 5 \cos \dfrac {3 \pi} 5$ $=$ $\displaystyle 1$ equating real parts $\displaystyle -4 \cos \dfrac \pi 5 \cos \paren {\dfrac \pi {10} + \dfrac \pi 2}$ $=$ $\displaystyle 1$ $\displaystyle -4 \cos \dfrac \pi 5 \paren {-\sin \dfrac \pi {10} }$ $=$ $\displaystyle 1$ $\displaystyle 4 \cos \dfrac \pi 5 \sin \dfrac \pi {10}$ $=$ $\displaystyle 1$

$\blacksquare$