Answer
$\dfrac{2\sqrt{3}}{3}$
Work Step by Step
$\sec{(\theta+ 2 \pi k}) = \sec{\theta}$
$\therefore \sec{\dfrac{25}\pi{6}} = \sec{\left(\dfrac{\pi}{6}+ (2 \pi \times 2) \right)} = \sec{\dfrac{\pi}{6}}$
$\sec{\dfrac{\pi}{6}} \\\\
= \dfrac{1}{\cos{\frac{\pi}{6}}}\\\\
=\dfrac{1}{\frac{\sqrt3}{2}}\\\\
=1 \cdot \dfrac{2}{\sqrt3}\\\\
=\dfrac{2}{\sqrt3}\\\\
=\boxed{\dfrac{2\sqrt{3}}{3}}$
