Complex Triangle inequality

Question

Given that

$|z_1| = |z_2| = 1 $

How would I use the triangle inequality to prove

$|z_1 + 1|+|z_2+1|+|z_1z_2+1|\geq|z_1+1|+|(z_2+1)-(z_1z_2+1)|$

And

$|z_2-z_1z_2| = |1-z_1|$

Answer 1

Notice that $|a|=|-a|$ and that $|b|+|a| \geq |b+a|$ so

$$|z_2+1|+|z_1z_2+1|\geq|(z_2+1)-(z_1z_2+1)|$$

And for the second $$|z_2-z_1z_2| = |z_2(1-z_1)|= |z_2||1-z_1| = 1\cdot |1-z_1|$$