Find the largest positive real number of $\delta $ such that $|\cos x - \cos y | < \sqrt 2$ whenever $|x-y| < \delta?$
a) $\sqrt 2$
b) $3/2$
c) $\pi/2$
d) $2$
My approach: $|\cos x - \cos y| =|-2 \sin\frac{x-y}{2}\sin\frac{x+y}{2}|<\sqrt 2$
$$ \implies |-2 \sin\frac{x-y}{2}\sin\frac{x+y}{2}| \leq|x-y|< \epsilon (=\delta)=\sqrt 2$$ as $|\sin x| \leq |x| \forall x \in \mathbb{R}$
So, $\epsilon=\delta = \sqrt 2 $ is one obvious possibility. but it is not the maximum.
I got stuck on how to reach the correct answer from here please give guidance.
Thanks in advanced