sin ⁡ π 8 = sin ⁡ 22.5 ∘ = 1 2 2 − 2 , {\displaystyle \sin {\frac {\pi }{8}}=\sin 22.5^{\circ }={\frac {1}{2}}{\sqrt {2-{\sqrt {2}}}},} cos ⁡ π 8 = cos ⁡ 22.5 ∘ = 1 2 2 + 2 {\displaystyle \cos {\frac {\pi }{8}}=\cos 22.5^{\circ }={\frac {1}{2}}{\sqrt {2+{\sqrt {2}}}}\,} tan ⁡ π 8 = tan ⁡ 22.5 ∘ = 2 − 1 {\displaystyle \tan {\frac {\pi }{8}}=\tan 22.5^{\circ }={\sqrt {2}}-1\,} cot ⁡ π 8 = cot ⁡ 22.5 ∘ = 2 + 1 = δ S {\displaystyle \cot {\frac {\pi }{8}}=\cot 22.5^{\circ }={\sqrt {2}}+1=\delta _{S}\,} , the silver ratio