Represente na forma trigonométrica

$$\begin{array}{*{20}{l}}
{\frac{{1 + \sqrt 2  + i}}{{1 + \sqrt 2  – i}}}& = &{\frac{{1 + \sqrt 2  + i}}{{1 + \sqrt 2  – i}} \times \frac{{1 + \sqrt 2  + i}}{{1 + \sqrt 2  + i}}} \\
{}& = &{\frac{{{{\left( {1 + \sqrt 2  + i} \right)}^2}}}{{{{\left( {1 + \sqrt 2 } \right)}^2} + 1}}} \\
{}& = &{\frac{{{{\left( {1 + \sqrt 2 } \right)}^2} + 2\left( {1 + \sqrt 2 } \right)i – 1}}{{1 + 2\sqrt 2  + 2 + 1}}} \\
{}& = &{\frac{{1 + 2\sqrt 2  + 2 + 2\left( {1 + \sqrt 2 } \right)i – 1}}{{4 + 2\sqrt 2 }}} \\
{}& = &{\frac{{2\left( {1 + \sqrt 2 } \right) + 2\left( {1 + \sqrt 2 } \right)i}}{{2\left( {2 + \sqrt 2 } \right)}}} \\
{}& = &{\frac{{\left( {1 + \sqrt 2 } \right) + \left( {1 + \sqrt 2 } \right)i}}{{\left( {2 + \sqrt 2 } \right)}} \times \frac{{2 – \sqrt 2 }}{{2 – \sqrt 2 }}} \\
{}& = &{\frac{{\left( {2 – \sqrt 2  + 2\sqrt 2  – 2} \right) + \left( {2 – \sqrt 2  + 2\sqrt 2  – 2} \right)i}}{{4 – 2}}} \\
{}& = &{\frac{{\sqrt 2 }}{2} + \frac{{\sqrt 2 }}{2}i} \\
{}& = &{\operatorname{cis} \frac{\pi }{4}}
\end{array}$$