If $x+iy=\sqrt\frac{1+i}{1-i}$, where $x$ and $y$ are real, prove that $x^2+y^2=1$

Question

If $x+iy=\sqrt\frac{1+i}{1-i}$, where $x$ and $y$ are real, prove that $x^2+y^2=1$

I tried multiplying $\sqrt{(\frac{1+i}{1-i})(\frac{1+i}{1+i})}=\sqrt{i}$ but I'm not sure what to do after

thanks in advance :)))

Answer 1

We need to prove that $$\left|\sqrt\frac{1+i}{1-i}\right|=1$$ or $$\sqrt{\left|\frac{1+i}{1-i}\right|}=1$$ or $$\sqrt1=1.$$ Done!

Answer 2

Let $z=x+iy$. Then $z^2=\frac{1+i}{1-i}$, hence $1=|z|^2$ and so $x^2+y^2=|z|^2=1$.

Answer 3

Just observe that numerator and denominator have the same modulus since $1+i$ and $1-i$ are conjugate, and the modulus is a multiplicative function.