{\cos ^3}\left( {{\pi \over 8}} \right)$$$${\cos}\left( {{3\pi \over 8}} \right)+{\sin ^3}\left( {{\pi \over 8}} \right)$$$${\sin}\left( {{3\pi \over 8}} \right) = ${\cos ^3}\left( {{\pi \over 8}} \right)\sin \left( {{\pi \over 8}} \right) + {\sin ^3}\left( {{\pi \over 8}} \right)\cos \left( {{\pi \over 8}} \right)$ = $\sin \left( {{\pi \over 8}} \right)\cos \left( {{\pi \over 8}} \right)\left[ {{{\cos }^2}\left( {{\pi \over 8}} \right) + {{\sin }^2}\left( {{\pi \over 8}} \right)} \right]$ = $\sin \left( {{\pi \over 8}} \right)\cos \left( {{\pi \over 8}} \right)$ $\times$ 1 = ${1 \over 2} \times 2\sin \left( {{\pi \over 8}} \right)\cos \left( {{\pi \over 8}} \right)$ = ${1 \over 2}\sin \left( {{\pi \over 4}} \right)$ = ${1 \over {2\sqrt 2 }}$