trignometry

let x=e^2Ï€i/11
x¹⁰=e^20Ï€i/11=cos(20Ï€/11)+isin(20Ï€/11)=cos(2Ï€/11)-isin(2Ï€/11)
x-x¹⁰=2isin(2Ï€/11)
4isin(2Ï€/11)=2(x-x¹⁰)

itan(3Ï€/11)=[cos(6Ï€/11)+isin(6Ï€/11)-1][cos(6Ï€/11)+isin(6Ï€/11)+1]=(x³-1)/(x³+1)=(x³-x³³)/(x³+1)
=x³-x⁶+x⁹-x¹²+x¹⁵ -x¹⁸+x²¹-x²⁴+x²⁷-x³⁰
=x³-x⁶+x⁹-x+x⁴ -x⁷+x¹⁰-x²+x⁵-x⁸
4isin(2Ï€/11)+itan(3Ï€/11)= x³-x⁶+x⁹-x+x⁴ -x⁷+x¹⁰-x²+x⁵-x⁸+ 2(x-x¹⁰)
=[x+x³+x⁴+x⁵+x⁹]- [x²+x⁶+x⁷+x⁸+x¹⁰]
S S'
so S=[x+x³+x⁴+x⁵+x⁹]
and S'=[x²+x⁶+x⁷+x⁸+x¹⁰] =[x^-1+x^-3+x^-4+x^-5+x^-9]

1+S+S'=1+x+x²+..+x¹⁰=(1-x¹¹)/(1-x)=0
so S+S'=-1
Find SS' which will be 5+2(S+S')=3
now S+S' is known and SS' is known so we can find S-S'
which is ±i√11

4isin(2Ï€/11)+itan(3Ï€/11)=±i√11
or 4sin(2Ï€/11)+tan(3Ï€/11)=√11 (it cannot be negative)