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

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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
\(\displaystyle -4 \cos \dfrac \pi 5 \cos \paren {\dfrac \pi {10} + \dfrac \pi 2}\) \(=\) \(\displaystyle 1\)
\(\displaystyle -4 \cos \dfrac \pi 5 \paren {-\sin \dfrac \pi {10} }\) \(=\) \(\displaystyle 1\)
\(\displaystyle 4 \cos \dfrac \pi 5 \sin \dfrac \pi {10}\) \(=\) \(\displaystyle 1\)

$\blacksquare$


Sources