$$\sqrt{4+\sqrt{4+\sqrt{4-\sqrt{4+\sqrt{4+\sqrt{4-...}}}}}} = 2(\cos{\frac{4\pi}{19}}+\cos{\frac{6\pi}{19}}+\cos{\frac{10\pi}{19}})$$
กำหนดให้ $\displaystyle{x = \sqrt{4+\sqrt{4+\sqrt{4-\sqrt{4+\sqrt{4+\sqrt{4-...}}}}}}}$
จะได้ $\displaystyle{x = \sqrt{4+\sqrt{4+\sqrt{4-x}}}}$ ทิ้งไว้เท่านี้ก่อน
กำหนด $\displaystyle{a_k = \cos{\frac{a\pi}{19}}}$
ดังนั้น จะได้ว่า
i.) $\displaystyle{4a_k^2 = 2+2a_k}$
ii.) $\displaystyle{8a_ka_n = 4a_{k+n}+4a_{k-n}}$
iii.) $\displaystyle{a_{19-k} = a_{19+k} = a_{-k} = -a{-k}}$
iv.) $\displaystyle{a_2+a_4+a_6+a_8+a_{10}+a_{12}+a_{14}+a_{16}+a_{18} = -\frac{1}{2}}$
กำหนดให้ $\displaystyle{P = \cos{\frac{2\pi}{19}}+\cos{\frac{4\pi}{19}}+ \cos{\frac{6\pi}{19}}+...+ \cos{\frac{18\pi}{19}}}$
คูณด้วย $\displaystyle{2\sin{\frac{\pi}{19}}}$ ทั้งสองข้าง และใช้สูตรผลคูณจะได้
$\displaystyle{2\sin{\frac{\pi}{19}}P = (\sin{\frac{3\pi}{19}}-\sin{\frac{\pi}{19}})+(\sin{\frac{5\pi}{19}}-\sin{\frac{3\pi}{19}})+(\sin{\frac{7\pi}{19}}-\sin{5\frac{\pi}{19}})+...+(\sin{\frac{19\pi}{19}}-\sin{\frac{17\pi}{19}})}$
$\displaystyle{2\sin{\frac{\pi}{19}}P = -\sin{\frac{\pi}{19}}}$
$P=-\frac{1}{2}$
กำหนดให้ $\displaystyle{x_1 = 2(a_4+a_6+a_{10})}$
$\displaystyle{x_1^2 - 4 = 4a_4^2+4a_6^2+4a_{10}^2+8a_4a_6+8a_4a_{10}+8a_6a_{10}-4}$
$\displaystyle{= 2a_8+2a_{12}+2a_{20}+4a_{10}+4a_{2}+4a_{14}+4a_6+4a_{16}+4a_4+2}$
$\displaystyle{= 2+4(a_2+a_4+a_6+a_8+a_{10}+a_{12}+a_{14}+a_{16}+a_{18})-2a_8-2a_{12}+2a_{20}-4a_{18}}$
$\displaystyle{= 2(a_1+a_7+a_{11})}$
กำหนดให้ $\displaystyle{x_2 = x_1^2-4 =2(a_1+a_7+a_{11})}$
$\displaystyle{x_2^2-4 = 4a_1^2+4a_7^2+4a_{11}^2+8a_1a_7+8a_1a_{11}+8a_7a_{11}-4}$
$\displaystyle{= 2a_2+2a_{14}+2a_{22}+4a_{6}+4a_{8}+4a_{10}+4a_{12}+4a_{4}+4a_{18}+2}$
$\displaystyle{= 2+4(a_2+a_4+a_6+a_8+a_{10}+a_{12}+a_{14}+a_{16}+a_{18})-2a_2-2a_{14}+2a_{16}-4a_{22}}$
$\displaystyle{= 2(a_3+a_5+a_{17})}$
กำหนดให้ $\displaystyle{x_3 = x_2^2-4 = 2(a_3+a_5+a_{17})}$
$\displaystyle{x_3^2-4 = 4a_3^2+4a_5^2+4a_{17}^2+8a_3a_5+8a_3a_{17}+8a_5a_{17}-4}$
$\displaystyle{= 2a_6+2a_{10}+2a_{34}+4a_{2}+4a_{8}+4a_{14}+4a_{20}+4a_{12}+4a_{22}+2}$
$\displaystyle{= 2a_6+2a_{10}+2a_{4}+4a_{2}+4a_{8}+4a_{14}+4a_{18}+4a_{12}+4a_{16}+2}$
$\displaystyle{= 2+4(a_2+a_4+a_6+a_8+a_{10}+a_{12}+a_{14}+a_{16}+a_{18})-2a_4-2a_{6}-2a_{10}}$
$\displaystyle{= -2(a_4+a_6+a_{10}) = -x_1}$
ดังนั้น $\displaystyle{((x_1^2-4)^2-4)^2-4 = -x_1}$
$\displaystyle{((x_1^2-4)^2-4)^2 = 4-x_1}$
$\displaystyle{x_1 = 2(a_4+a_6+a_{10}) \approx 2.51 < 4}$ สามารถใส่รูทได้
$\displaystyle{(x_1^2-4)^2-4 = \sqrt{4-x_1}}$
$\displaystyle{(x_1^2-4)^2=4+\sqrt{4-x_1}}$
$\displaystyle{x_1^2-4 = \sqrt{4+\sqrt{4-x_1}}}$
$\displaystyle{x_1 = \sqrt{4+\sqrt{4+\sqrt{4-x_1}}}}$
เมื่อแทนค่า $x_1$ กลับเข้าไปใน square root ก็จะได้ค่าตามโจทย์ตามที่่ต้องการ