complex numbers - For $zinmathbb{C}, |z|=1$, Prove that $Big(frac{1+ia}{1-ia}Big)^4=z$ has all roots real and distinct

Given $z\in\mathbb{C}$ with $|z|=1$, then prove that the equation $\Big(\dfrac{1+ia}{1-ia}\Big)^4=z$ has all roots real and distinct

My Attempt
$$
z=e^{i\theta}\implies z^{1/4}=e^{i\theta/4}=\frac{1+ia}{1-ia}.1^{1/4}=\frac{1+ia}{1-ia}.(1,-1,i,-i)
$$
Is it correct to introduce the quadratic root of unity for the complete solution ?

Case 1: $1^{1/4}=1$
$$
\frac{1}{ia}=\frac{e^{i\theta/4}+1}{e^{i\theta/4}-1}=\frac{2\cos^2\theta/8+2i\sin\theta/8.\cos\theta/8}{-2\sin^2\theta/8+2i\sin\theta/8.\cos\theta/8}\\

\frac{1}{ia}=i\cot\theta/8\\\boxed{a=-\tan\theta/8}
$$
Case 2: $1^{1/4}=-1$
$$
\frac{1}{ia}=\frac{-e^{i\theta/4}+1}{-e^{i\theta/4}-1}=\frac{2\sin^2\theta/8-2i\sin\theta/8.\cos\theta/8}{-2\cos^2\theta/8-2i\sin\theta/8.\cos\theta/8}\\
\frac{1}{ia}=i\tan\theta/8\implies\boxed{a=-\cot\theta/8}
$$
Case 3: $1^{1/4}=-i$
$$
\frac{1}{ia}=\frac{ie^{i\theta/4}+1}{ie^{i\theta/4}-1}=\frac{1-\sin\theta/4+i\cos\theta/4}{-1-\sin\theta/4+i\cos\theta/4}\\

=\frac{1-\cos(\pi/2-\theta/4)+i\sin(\pi/2-\theta/4)}{-1-\cos(\pi/2-\theta/4)+i\sin(\pi/2-\theta/4)}\\
=\frac{2\sin^2(\pi/4-\theta/8)+2i\sin(\pi/4-\theta/8)\cos(\pi/4-\theta/8)}{-2\cos^2(\pi/4-\theta/8)+2i\sin(\pi/4-\theta/8)\cos(\pi/4-\theta/8)}\\
\frac{1}{ia}=-i\tan(\pi/4-\theta/8)\\
\boxed{a=\cot(\pi/4-\theta/8)=\tan(\pi/4+\theta/8)}
$$
Case 4: $1^{1/4}=i$
$$
\frac{1}{ia}=\frac{-ie^{i\theta/4}+1}{-ie^{i\theta/4}-1}=\frac{1+\sin\theta/4-i\cos\theta/4}{-1+\sin\theta/4-i\cos\theta/4}\\
=\frac{1+\cos(\pi/2-\theta/4)-i\sin(\pi/2-\theta/4)}{-1+\cos(\pi/2-\theta/4)-i\sin(\pi/2-\theta/4)}\\
=\frac{2\cos^2(\pi/4-\theta/8)-2i\sin(\pi/4-\theta/8)\cos(\pi/4-\theta/8)}{-2\sin^2(\pi/4-\theta/8)-2i\sin(\pi/4-\theta/8)\cos(\pi/4-\theta/8)}\\

=i.\cot(\pi/4-\theta/8)\\
\frac{1}{ia}=i\tan(\pi/4+\theta/8)\\
\boxed{a=-\cot(\pi/4+\theta/8)}
$$
Is it the right way to approach the problem ? and whats the easiest way to solve this ?

Answer

For the reality of the roots, take module on both sides :
$$\left|\frac{i-a}{-i-a}\right|=|z|^{1/4}=1$$
so if $A$ is the point of affix $a$, $I$ of affix $i$ and $I'$ of affix $-i$, this says that $AI=AI'$, so $A$ belongs to the perpendicular bisector of segment $[II']$, therefore the real axis.

For distinct roots, take argument : $(\overrightarrow{AI'};\overrightarrow{AI})=\frac14\arg(z)+k\frac\pi2$ with $k\in\{0,1,2,3\}$, this means 4 distinct points.

(There could be a better argument for last point).

With some calculus (which doesn't explain why roots are real) : let $z=e^{i\theta}$, and for $k\in\{0,1,2,3\}$, $\theta_k=\theta/4+k\pi/2$. Equation solves as
$$a=\frac{e^{i\theta_k}-1}{i(e^{i\theta_k}+1)} = \frac{e^{i\theta_k/2}-e^{-i\theta_k/2}}{i(e^{i\theta_k/2}+e^{-i\theta_k/2})} = \tan(\theta_k/2) = \tan(\theta/8+k\pi/4)$$
OK, so it's real, but why ?