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Find the exact value of cos pi/16

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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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