Show that $|b-a|\geq|\cos a-\cos b|$ for all real numbers $\,a\,$ and $\,b$

Even faster you can use fundamental theorem of calculus and assuming for example $a≤b$ $$ \begin{align*} |\cos(a)-\cos(b)| &= \left|\int_a^b \sin(x)\,\mathrm{d}x\right| ≤ \int_a^b \left|\sin(x)\right|\mathrm{d}x \\ &≤ \int_a^b 1\,\mathrm{d}x = |b-a| \end{align*} $$