Veja o triângulo (está em anexo)
[tex]tg \beta =AC/AB=1 / (2 + \sqrt{3} )[/tex]
[tex]tg \alpha =AC/AD=1/1=1[/tex]
[tex]tg ( \alpha + \beta )=(tg \alpha +tg \beta )/(1-tg \alpha *tg \beta )[/tex]
[tex]tg( \alpha + \beta )=(1 + [1/(2+ \sqrt{3} )])/(1-1*[1/(2+ \sqrt{3}])[/tex]
[tex]tg(\alpha+\beta)= (1 + [1/(2+\sqrt{3})/(1-[1/(2+\sqrt{3})])[/tex]
[tex]tg(\alpha+\beta)=[(2+\sqrt{3}+1)/(2+\sqrt{3})]/[(2+\sqrt{3}-1)/(2+\sqrt{3} )][/tex]
[tex]tg(\alpha+\beta)=[(3+\sqrt{3})/(2+\sqrt{3})]/[(1+ \sqrt{3})/(2+\sqrt{3})] [/tex]
[tex]tg(\alpha+\beta)=[(3+\sqrt{3})/(2+\sqrt{3})]*(2+\sqrt{3})/(1+ \sqrt{3})[/tex]
[tex]tg( \alpha + \beta )=(3+ \sqrt{3})/(1+ \sqrt{3})[/tex]
[tex]tg( \alpha + \beta )=(3+ \sqrt{3})*(1- \sqrt{3})/[(1+ \sqrt{3})*(1- \sqrt{3} )][/tex]
[tex]tg( \alpha + \beta)=(3-3 \sqrt{3} + \sqrt{3} - \sqrt{3}^{2})/(1^{2} - \sqrt{3}^{2})[/tex]
[tex]tg( \alpha + \beta )=(3-2 \sqrt{3}-3)/(1-3)[/tex]
[tex]tg( \alpha + \beta )=-2 \sqrt{3} /(-2)[/tex]
[tex]tg( \alpha + \beta )= \sqrt{3} [/tex]
