<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] Add and subtract $\cos\left(\frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right)$ in the numerator to get:
$$\lim_{\theta \to \dfrac {\pi}{4}} \dfrac {\cos \theta - \sin \theta}{\theta - \dfrac {\pi}{4}} = \lim_{\theta \to \dfrac {\pi}{4}} \dfrac {\cos \theta - \cos\left(\frac{\pi}{4}\right) - \left(\sin \theta - \sin\left(\frac{\pi}{4}\right)\right)}{\theta - \dfrac {\pi}{4}} $$
$$= \lim_{\theta \to \dfrac {\pi}{4}} \dfrac {\cos \theta - \cos\left(\frac{\pi}{4}\right)}{\theta - \dfrac {\pi}{4}} - \lim_{\theta \to \dfrac {\pi}{4}} \dfrac {\sin \theta - \sin\left(\frac{\pi}{4}\right)}{\theta - \dfrac {\pi}{4}} = -\sin\left(\dfrac {\pi}{4}\right) - \cos\left(\dfrac {\pi}{4}\right) = -\sqrt{2}$$
where we used the definition for derivatives of sine and cosine.
</s>