Answer
$1$
Work Step by Step
Recall:
$\cos{(\theta+360^{\circ})} = \cos{\theta}$
This means that
$\cos{400^{\circ}} = \cos{(40^{\circ}+360^{\circ})} = \cos{40^{\circ}}$
Thus,
$\cos{400^{\circ}} \cdot \sec{40^{\circ}} = \cos{40^{\circ}}\cdot \sec{40^{\circ}}$
Since $\sec{\theta} = \dfrac{1}{\cos{\theta}}$, then
$\sec{40^{\circ}} = \dfrac{1}{\cos{40^{\circ}}}$
Hence,
$\cos{40^{\circ}}\cdot \sec{40^{\circ}} = \cos{40^{\circ}}\cdot \dfrac{1}{\cos{40^{\circ}} }= \boxed{1}$