Chapter 7 - Trigonometric Identities and Equations - 7.4 Double-Angle and Half-Angle Identities - 7.4 Exercises - Page 693: 31

$-\cos \left(\dfrac {4\pi }{5}\right) =\cos \left( \dfrac {\pi }{5}\right) $

$\sin ^{2}\alpha -\cos ^{2}\alpha =-\cos ^{2}\alpha \Rightarrow \sin ^{2}\dfrac {2\pi }{5}-\cos ^{2}\dfrac {2\pi }{5}=-\cos \dfrac {4\pi }{5}=\cos \left( \pi -\dfrac {4\pi }{5}\right) =\cos \left( \dfrac {\pi }{5}\right) $