$\left(1+\sec {20}^0\right)\left(1+\sec {40}^0\right)\left(1+\sec {80}^0\right)=$

$\begin{array}{rcl} & & \left(1+\sec {20}^0\right) \left(1+\sec {40}^0\right) \left(1+\sec {80}^0\right)\\ & =& \frac{\left(1+\cos {20}^0\right)}{\cos {20}^0} \frac{\left(1+\cos {40}^0\right)}{\cos {40}^0} \frac{\left(1+\cos {80}^0\right)}{\cos {80}^0}\\ & =& \frac{2{\cos }^2 {10}^0 2{\cos }^2 {20}^0 2{\cos }^2 {40}^0}{\cos {20}^0 \cos {40}^0 \cos {80}^0}\\ & =& 8\frac{{\sin }^2 80}{64{\sin }^2 {10}^0 {\displaystyle \frac{1}{4}}\cos {60}^0}\\ & =& \frac{{\cos }^2 {10}^0}{2{\sin }^2 {10}^0 {\displaystyle \frac{1}{2}}}\\ & =& {\cot }^2 {10}^0\end{array}$