Mathematical Induction with tan extremely hard

Question

Extremely hard high school mathematical induction question. Part i is easy. Part ii not so sure but I'm not sure how to get rid of the pi/2 when tan pi/2 is non existent. I think multiplying that given equation by 2 may work? and then creating several fractions such as $$ {x+y} \over {1-xy} $$ $$ {x+z} \over {1-xz} $$ $$ {z+y} \over {1-zy} $$

THen for part iii.... absolutely no clue

Answer 1

Hint:

Use $\arctan x+\arctan y=\dfrac{\pi}{2}-\arctan z$ and with $w_n=\prod_k (1-ix_k)$ then $$\arg w_n=\arg\prod_k (1-ix_k)=\sum_k \arg(1-ix_k)$$