Answer
$2 \sqrt 2+\dfrac{4 \sqrt 3}{3}$
Work Step by Step
Note that $\sec x=\dfrac{1}{\cos x}$ and $\cot x=\dfrac{1}{\tan x}$.
We use the unit circle values to simplify.
Since $\cos{\frac{\pi}{4}}=\frac{\sqrt2}{2}$ and $\tan{\frac{\pi}{3}}=\sqrt3$, then:
$2 \sec \frac{\pi}{4} +4 \cot \frac{\pi}{3}\\
=2\cdot \dfrac{1}{\cos{\frac{\pi}{4}}} +4\cdot \dfrac{1}{\tan{\frac{\pi}{3}}}\\
= \dfrac{2}{ \sin \frac{\pi}{4}} + \dfrac{4}{ \tan (\pi/3)} \\
= \dfrac{2}{\frac{\sqrt2}{2}} + \dfrac{4}{ \sqrt3} \\
= \dfrac{4}{\sqrt2} + \dfrac{4 \sqrt 3}{3} \\
= \dfrac{4\sqrt2}{2} + \dfrac{4 \sqrt 3}{3} \\
=2 \sqrt 2+\dfrac{4 \sqrt 3}{3}$
