Pf: From problem definition/assumed finite k we have \(\tau > 0\). Additionally:
\(k=\lceil 2\pi/ \tau \rceil > 8 \implies 2\pi/ \tau > 7 \iff \tau < 2\pi/7\)
\(\square\)
Pf: \(\frac{d(\cos x - \sin x)}{dx}=-(\sin x + \cos x) < 0\) for \(x\in (0, \pi/4)\) therefore \(\cos x - \sin x\) is monotone decreasing on this domain. As such, \(\frac{1}{\cos x - \sin x}\) is monotone increasing \( \square\).
\[\begin{aligned} t\cdot|uv| &\geq |uw| + t\cdot|wv| \\ \iff t &\geq \frac{|uw|}{|uv|-|wv|} \\\\ LHS &\geq \frac{1}{\cos \tau - \sin \tau} \\ &\geq \frac{1}{\cos \angle_{vuw} - \sin \angle_{vuw} } \qquad \text{as $\angle_{vuw} \in (0,\tau]$ (1)(2)}\\ &=\frac{|uv|}{|uw|-|wv|}\\ &\geq\frac{|uw|}{|uv|-|wv|} \qquad \text{as $|uv| > |uw|$}\\ &=RHS \ \ \square\end{aligned}\]