How to find the value of $4\cos(\frac{\pi}{26})+\tan(\frac{2\pi}{13})$

Question

I have found in wolfram alpha that $\displaystyle 4\cos\left(\frac{\pi}{26}\right)+\tan\left(\frac{2\pi}{13}\right)=\sqrt{13+2\sqrt{13}}$.

How to prove this identity ? Thank you.

Answer 1

This is old problem can see: How prove this $\tan{\frac{2\pi}{13}}+4\sin{\frac{6\pi}{13}}=\sqrt{13+2\sqrt{13}}$

show that: The follow nice trigonometry

$$\tan{\dfrac{2\pi}{13}}+4\sin{\dfrac{6\pi}{13}}=\sqrt{13+2\sqrt{13}}$$

Answer 2

Well you can look here

http://mathworld.wolfram.com/TrigonometryAnglesPi13.html

there something is explained...