complex numbers, polar form

[Vali]

I have these complex numbers: $\displaystyle z_1=\sin(a)-\cos(a)+i(\sin(a)+\cos(a))$ and $\displaystyle z_2=\sin(a)+\cos(a)+i(\sin(a)-\cos(a))$

I need to find $\displaystyle a$ values such that the absolute value of complex number $\displaystyle w=(z_1)^n+(z_2)^n$ is maximum.The right answer is $\displaystyle \frac{k\pi}{n}+\frac{\pi}{2}$

My try: I wrote $\displaystyle (\sin(a)+\cos(a)=\sqrt{2}\sin(\frac{\pi}{4}+a)$ and $\displaystyle \sin(a)-\cos(a)=\sqrt{2}\cos(\frac{3\pi}{4}-a)$ so $\displaystyle z_1=\sqrt{2}[\cos(\frac{3\pi}{4}-a)+i\sin(\frac{pi}{4}+a)]$ but the polar form it's not good.How to convert it to the polar form ?How to approach this exercise?