Sum, Difference, and Multiple-Angle Formulas · Superchallenge problem

Find the exact value of $\dfrac{1}{2}\tan\dfrac{\pi}{6} + \dfrac{1}{4}\tan\dfrac{\pi}{12} + \dfrac{1}{8}\tan\dfrac{\pi}{24} - \dfrac{1}{8}\cot\dfrac{\pi}{24}$.