Stuck on Rudin's Statement on the Zeros of Cosine Function

Question

For cosine and sine function defined as

How do we derive the last inequality of (50)

from (48)

and (47) ?

Answer 1

From (47) and (48), if $x$ is real, $$ C(x)^2 + S(x)^2 = 1, $$ therefore $$ \left\lvert{C(x)}\right\rvert \leqslant 1, $$ i.e. $$ -1 \leqslant C(x) \leqslant 1. $$ Similarly, if $y$ is real, $$ -1 \leqslant C(y) \leqslant 1. $$ Therefore, if $x$ and $y$ are real, $$ -2 \leqslant C(x) - C(y) \leqslant 2. $$