Here is the issue and here is the code.
\begin{align*}
\left ( \frac{1+\sqrt{3}}{2\sqrt{2}}+ i \frac{\sqrt{3}-1}{2\sqrt{2}} \right )^{72} &= \left [ \left ( \frac{1}{2}\cdot \frac{\sqrt{2}}{2}+ \frac{\sqrt{3}}{2}\cdot \frac{\sqrt{2}}{2} \right )+ i \left ( \frac{\sqrt{3}}{2}\cdot \frac{\sqrt{2}}{2} -\frac{1}{2}\frac{\sqrt{2}}{2} \right ) \right ]^{72}\\
&=\left [ \left ( \cos \frac{\pi}{3}\cos \frac{\pi}{4} + \sin \frac{\pi}{3}\sin \frac{\pi}{4} \right ) +i \left ( \sin \frac{\pi}{3} \cos \frac{\pi}{4} - \cos \frac{\pi}{3} \sin \frac{\pi}{4} \right ) \right ]^{72} \\
&= \left [ \cos \left ( \frac{\pi}{3} - \frac{\pi}{4} \right ) +i \sin \left ( \frac{\pi}{3} - \frac{\pi}{4} \right )\right ]^{72}\\
&= \left ( \cos \frac{\pi}{12} + i \sin \frac{\pi}{12} \right )^{72}\\
&= \left ( \cos^2 \frac{\pi}{12} - \sin^2 \frac{\pi}{12} +2i \cos \frac{\pi}{12} \sin \frac{\pi}{12} \right )^{36}\\
&=\left ( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right )^{36} \\
&= \left ( \frac{\sqrt{3}}{2}+\frac{i}{2} \right )^{36} \\
&=\left [ \left ( \frac{\sqrt{3}}{2}+ \frac{i}{2} \right )^3 \right ]^{12} \\
&=1
\end{align*}
The line
\left [ \left ( \cos \frac{\pi}{3}\cos \frac{\pi}{4} + \sin \frac{\pi}{3}\sin \frac{\pi}{4} \right ) +i \left ( \sin \frac{\pi}{3} \cos \frac{\pi}{4} - \cos \frac{\pi}{3} \sin \frac{\pi}{4} \right ) \right ]^{72}
however exceeds the margins of the page so I want to break it. I used \\ and then on the new line & plus two \qquad to bring it in a favor position and altough it gets a result the compile ends up with errors which is something logical. The output is that the final ] in that line is not depicted at all.
How can I break up that line? Need I use alignat package?
\bigg[and\bigg]instead of\left[and\right]. Should produce what you want. There's also\big[\big]if you want slightly smaller brackets.\biggl[and\biggr]to get correct open and close spacing.