Chapter 7 - Trigonometric Identities and Equations - 7.4 Double-Angle and Half-Angle Identities - 7.4 Exercises - Page 693: 26

$\dfrac {\cos \dfrac {\pi }{4}}{2}=\dfrac {\sqrt {2}}{4}$

$2\cos ^{2}\alpha -1=\cos ^{2}\alpha \Rightarrow \cos ^{2}\dfrac {\pi }{8}-\dfrac {1}{2}=\dfrac {2\cos ^{2}\dfrac {\pi }{8}-1}{2}=\dfrac {\cos \left( 2\times \dfrac {\pi }{8}\right) }{2}=\dfrac {\cos \dfrac {\pi }{4}}{2}=\dfrac {\sqrt {2}}{4}$