Answer
$-\sqrt{2}$
Work Step by Step
$\cos{A} = \dfrac{1}{\sec{A}} = \dfrac{\sqrt{5}}{5}$
$\sin{A} = \sqrt{1-(\dfrac{\sqrt{5}}{5})^2 } = \dfrac{2\sqrt{5}}{5}$
$\cos{B} = \dfrac{1}{\sec{B}} = \dfrac{\sqrt{10}}{10}$
$\sin{B} = \sqrt{1-(\dfrac{\sqrt{10}}{10})^2 } = \dfrac{3\sqrt{10}}{10}$
$\cos{(A+B)} = \cos{A} \cos{B} - \sin{A} \sin{B}$
$\cos{(A+B)} = (\dfrac{\sqrt{5}}{5})(\dfrac{\sqrt{10}}{10}) - (\dfrac{2 \sqrt{5}}{5})(\dfrac{3 \sqrt{10}}{10})$
$\cos{(A+B)} = -\dfrac{\sqrt{2}}{2}$
$\sec{(A+B)} = \dfrac{1}{\cos{(A+B)} } = -\sqrt{2}$
