Given two complex numbers $z,w$ such that $ |z|=|w|=1$. Find the correct statement.

Given two complex numbers $z,w$ with unit modulus (i.e., $ |z|=|w|=1$), which of the following statements will always be correct?

a.) $|z+w|ltsqrt2$ and $|z-w|ltsqrt2$

b.) $|z+w|lesqrt2$ and $|z-w|gesqrt2$

c.) $|z+w|gesqrt2$ or $|z-w|gesqrt2$

d.) $|z+w|ltsqrt2$ or $|z-w|ltsqrt2$

It is a multiple choice question and only one option is correct.

My approach: As modulus of $z$ and $w$ is 1. Let $z=e^{ialpha_1}$ and $w=e^{ialpha_2}$. Now,

$$|z+w|= |e^{ialpha_1}+e^{ialpha_2}|$$

$$|z+w|=|2 cos(frac{alpha_1-alpha_2}{2})e^{frac{i(alpha_1+alpha_2)}{2}}|$$

$$|z+w|=2 |cos(frac{alpha_1-alpha_2}{2})|$$

I don’t know how to proceed after this. Any help would be appreciated.