Answer     $1$

The goal is to convert it to an expression with respective to $\sin\frac{\pi}{7}$.

The first term can be first transformed using double angle formula as $$4\cos\frac{2\pi}{7}\cos\frac{\pi}{7}=\frac{1}{\sin\frac{\pi}{7}}\left(4\sin\frac{\pi}{7}\cos\frac{2\pi}{7}\cos\frac{\pi}{7}\right)=\frac{\sin\frac{4\pi}{7}}{\sin\frac{\pi}{7}}=\frac{\sin\frac{3\pi}{7}}{\sin\frac{\pi}{7}}$$

Applying the triple angel formula, we have $$\frac{3\sin\frac{\pi}{7}-4\sin^3(\frac{\pi}{7})}{\sin\frac{\pi}{7}}=3-4\sin^2\frac{\pi}{7}$$

The second term can be rewritten as $$2\cos\frac{2\pi}{7} = 2\left(1-2\sin^2\frac{\pi}{7}\right)$$

Hence, the final answer is $1$.