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Prove that tanpi/(16)=sqrt(4+2sqrt(2))-(...

Prove that `tanpi/(16)=sqrt(4+2sqrt(2))-(sqrt(2)+1)`

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`tan ""(pi)/(16)=tan 11.25^(@)`
We know that on `22.5^(@)=sqrt(2)-1`
`tan 2theta=(2tan theta)/(1-tan^(2)theta)`
Put `theta=11.5^(@)`
`therefore tan 22.5^(@)=(2 tan 11.25^(@))/(1-tan^(2)11.25^(@))`
`therefore (sqrt(2)-1)x^(2)+2x-(sqrt(2)-1)=0`, where `x=tan 11.25^(@)`
`therefore x=(-2+sqrt(4+4(sqrt(2)-1))^(2))/(2(sqrt(2)-1))`
`(-1)/(sqrt(2)-1)+(sqrt(4-2sqrt(2)))/(sqrt(2)-1)`
`=-(sqrt(2)+1)+sqrt(4-2sqrt(2)).(sqrt(2)+1)`
`=-(sqrt(2)+1)+sqrt((4-2sqrt(2))(sqrt(2)+1)^(2))`
`=-(sqrt(2)+1)+sqrt((4-2sqrt(2))(3+2sqrt(2)))`
`=-(sqrt(2)+1)+sqrt(4+2sqrt(2))`
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