Find the exact value of `cospi/16`
.
We use a half-angle formula: `cos (alpha/2)=sqrt(1/2(1+cos
alpha))`
(1) `cos pi/8=cos(pi/4)/2=sqrt(1/2(1+cos pi/4))`
`=sqrt(1/2(1+sqrt(2)/2))`
`=sqrt((2+sqrt(2))/4)`
`=sqrt(2+sqrt(2))/2`
(2) `cos pi/16 = cos (pi/8)/2`
`=sqrt(1/2(1+cos pi/8))`
`=sqrt(1/2((sqrt(2+sqrt(2)))/2))`
`=sqrt((2+sqrt(2+sqrt(2)))/4)`
`=(sqrt(2+sqrt(2+sqrt(2))))/2`
Thus
`cos pi/16 = (sqrt(2+sqrt(2+sqrt(2))))/2~~.9807852804`
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