Find the sub-intervals of
\[
(0,\pi)
\]
in which
\[
f(x)=\tan^{-1}(\sin x-\cos x)
\]
is increasing and decreasing.
Show Hint
Whenever you encounter linear combinations of sine and cosine functions like \(a\sin x + b\cos x\), convert them into a single sine or cosine function using the transformation factor \(\sqrt{a^2 + b^2}\) to simplify your inequality analysis.