3.108: Equilateral triangle and sec identity

The diagram shows an equilateral triangle $ABC$ with side $a$. $M$ is the midpoint of $AC$ and angle $AMB=90^{\circ}$. Use the diagram to find $\mathrm{sec}\,30^{\circ}$.

Show that $\dfrac{1}{\mathrm{sec}\,x-1}+\dfrac{1}{\mathrm{sec}\,x+1}$ can be written as $2\,\mathrm{cosec}\,x\,\mathrm{cot}\,x$.