<s>[INST] prove that $\int_0^\frac{\pi}{4} \frac{1-\cos x}{x} \,dx < \frac{\pi^2}{64}$ prove that $\int_0^\frac{\pi}{4} \frac{1-\cos x}{x} \,dx <  \frac{\pi^2}{64}$
I showed that in 
$$ \forall x \in [0,\frac{\pi}{4}] \quad  \int_0^\frac{\pi}{4} \frac{1-\cos x}{x} \,dx \le \int_0^\frac{\pi}{4} \frac{1-\cos(\frac{\pi}{4})}{\frac{\pi}{4}} \,dx = 1 -\frac{\sqrt2}{2} $$
but it's not good enough
 [/INST] Using the fact that $\sin(x)\le x$ for $x\ge0$,
$$
\begin{align}
\int_0^{\pi/4}\frac{1-\cos(x)}x\,\mathrm{d}x
&=\int_0^{\pi/4}\frac{2\sin^2(x/2)}x\,\mathrm{d}x\\
&=\int_0^{\pi/4}\frac{\sin(x/2)}{x/2}\cdot\sin(x/2)\,\mathrm{d}x\\
&\le\int_0^{\pi/4}1\cdot\frac x2\,\mathrm{d}x\\[4pt]
&=\frac{\pi^2}{64}
\end{align}
$$
</s>