<s>[INST] Evaluate: $\lim_{\theta \to \frac {\pi}{4}} \dfrac {\cos \theta - \sin \theta}{\theta - \frac {\pi}{4}}$ 
Evaluate: $\lim_{\theta \to \frac {\pi}{4}} \dfrac {\cos \theta - \sin \theta}{\theta - \dfrac {\pi}{4}}$.

My Attempt:
\begin{align}
\lim_{\theta \to \frac {\pi}{4}} \dfrac {\cos \theta - \sin \theta }{\theta - \dfrac {\pi}{4}}
&=\lim_{\theta \to \frac {\pi}{4}} \dfrac {\cos \theta - \cos \dfrac {\pi}{4} + \sin \dfrac {\pi}{4} - \sin \theta}{\theta - \dfrac {\pi}{4}}
\\
&=\lim_{\theta \to \frac {\pi}{4}} \dfrac {2\sin \dfrac {\pi-4\theta }{8}\cos \dfrac {\pi+4\theta}{8} - 2\sin \dfrac {4\theta + \pi}{8}\sin \dfrac {4\theta -\pi}{8}}{\theta - \dfrac {\pi}{4}}.
\end{align}
How do I proceed?
 [/INST] $$\lim_{\theta \to \frac {\pi}{4}} \frac {\cos \theta - \sin \theta}{\theta - \frac {\pi}{4}} $$
Let $x:=\theta-\pi /4$, the limit you want to find equals to
$$\lim_{x \to 0} \frac {\cos (x+\pi /4) - \sin (x+\pi /4)}{x} $$
$$=\lim_{x \to 0} -\frac {\sqrt 2(\sin (x+\pi /4)\cos (\pi /4)-\sin (\pi /4)\cos (x+\pi /4))}{x} $$
As something we (intentionally) got in the bracket is the compound angle formula for $\sin$, we have the above limit to equal
$$\lim_{x \to 0} -\frac {\sqrt 2(\sin (x+\pi /4-\pi /4))}{x} $$
$$=\lim_{x \to 0} -\frac {\sqrt 2\sin{x}}{x} $$
$$=-\sqrt 2$$
</s>