Cosine of 72 Degrees/Proof 2

Theorem

$\cos 72 \degrees = \cos \dfrac {2 \pi} 5 = \dfrac {\sqrt 5 - 1} 4$

$\ds \cos 72 \degrees$

$=$

$\ds 2 \cos 36 \degrees - 1$

$\ds $

$=$

$\ds 2 \paren {\dfrac \phi 2}^2 - 1$

$\ds $

$=$

$\ds \dfrac {\phi^2} 2 - 1$

$\ds $

$=$

$\ds \dfrac {\phi + 1} 2 - 1$

$\ds $

$=$

$\ds \dfrac {\phi - 1} 2$

$\ds $

$=$

$\ds -\dfrac {1 - \phi} 2$

$\ds $

$=$

$\ds \dfrac {\phi^{-1} } 2$

$\ds $

$=$

$\ds \dfrac {\sqrt 5 - 1} 4$

Definition 2 of Golden Mean, and algebra

$\blacksquare$

1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.8$: Fibonacci Numbers: Exercise $19$