if a+ib=root(1+i)/root(1-i), then prove that a²+b²=1

Dear Student,

Please find below the solution to the asked query:
$$\begin{gathered} \Rightarrow \mathrm{a}+\mathrm{ib}=\frac{\sqrt{1+\mathrm{i}}}{\sqrt{1-\mathrm{i}}} \\ \mathrm{RHS} \\ \Rightarrow \frac{\sqrt{1+\mathrm{i}}}{\sqrt{1-\mathrm{i}}} \\ \mathrm{Multilply} \mathrm{and} \mathrm{divide} \mathrm{by} \sqrt{1-\mathrm{i}} \\ \Rightarrow \frac{\sqrt{1+\mathrm{i}}}{\sqrt{1-\mathrm{i}}}\times \frac{\sqrt{1+\mathrm{i}}}{\sqrt{1+\mathrm{i}}} \\ \Rightarrow \frac{1+\mathrm{i}}{\sqrt{1+1}}=\frac{1}{\sqrt{2}}+\mathrm{i}\frac{1}{\sqrt{2}} \\ \mathrm{Now} \mathrm{compare} \mathrm{with} \mathrm{LHS} \\ \mathrm{a}=\mathrm{b}=\frac{1}{\sqrt{2}} \\ \mathrm{So} \\ \Rightarrow {\mathrm{a}}^2+{\mathrm{b}}^2 \\ \Rightarrow {\left(\frac{1}{\sqrt{2}}\right)}^2+{\left(\frac{1}{\sqrt{2}}\right)}^2 \\ \Rightarrow \frac{1}{2}+\frac{1}{2} \\ \Rightarrow 1 \mathrm{Hence} \mathrm{Proved} \end{gathered}$$

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