de Moivre's identity states that
e^(ix)=cos(ix)+isin(ix)eix=cos(ix)+isin(ix) so taking
t_1=1-4sin^2theta =(1+2sintheta)(1-2sintheta) = (1-(e^(itheta)-e^(-itheta))/i) (1+(e^(itheta)-e^(-itheta))/i)t1=1−4sin2θ=(1+2sinθ)(1−2sinθ)=(1−eiθ−e−iθi)(1+eiθ−e−iθi)
t_2=3-4sin^2theta=(sqrt(3)+2sintheta)(sqrt(3)-2sintheta)= (sqrt(3)-(e^(itheta)-e^(-itheta))/i) (sqrt(3)+(e^(itheta)-e^(-itheta))/i)t2=3−4sin2θ=(√3+2sinθ)(√3−2sinθ)=(√3−eiθ−e−iθi)(√3+eiθ−e−iθi)
and
t_3=(e^(i2theta)-e^(-i2theta))/(2i)t3=ei2θ−e−i2θ2i
we have
t_1=e^(2itheta)+e^(-2itheta)-1t1=e2iθ+e−2iθ−1
t_2=e^(2itheta)+e^(-2itheta)+1t2=e2iθ+e−2iθ+1
and after multilying t_1t_2t_3t1t2t3
t_1t_2t_3=(e^(6itheta)-e^(-6itheta))/(2i) = sin(6theta)t1t2t3=e6iθ−e−6iθ2i=sin(6θ)