<s>[INST] Complex Numbers Questions... I am having trouble with these problems:


*

*Find all complex numbers $z$ satisfying the equation
$$\frac{z+1}{z-1} = i.$$

*The value
$$\left(\frac{1+\sqrt 3}{2\sqrt 2}+\frac{\sqrt 3-1}{2\sqrt 2}i\right)^{72}$$
is a positive real number. What real number is it?
On 1) , I was thinking about substituting $z$ for $a+bi$ and then solving. Is this correct?
 [/INST] For the first:   
We have $$i = 1 + \frac{2}{z-1} \Rightarrow z= -i$$    
For the second:   
We can write $$\frac{1+\sqrt{3}}{2\sqrt{2}} = \frac{1}{2}\frac{1}{\sqrt{2}} + \frac{\sqrt{3}}{2}\frac{1}{\sqrt{2}} = \cos 60^\circ \cos 45^\circ +\sin 60^\circ \sin 45^\circ = \cos (60^\circ -45^\circ ) = \cos 15^\circ$$ Similarly, we have, $\frac{\sqrt{3}-1}{2\sqrt{2}} = \sin 15^\circ$. Thus, using De Moivre's formula, we get, $$(\cos 15^\circ + \sin 15^\circ )^{72} = (\cos 1080^\circ + \sin 1080^\circ) = (\cos 6\pi + \sin 6\pi) = 1$$   Hope it helps.
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