Answer
$(2-\sqrt 2)\sqrt {2+\sqrt 2}$
Work Step by Step
Step 1. $cos(\frac{15\pi}{8})=\sqrt {\frac{1+cos(\frac{15\pi}{4})}{2}}=\sqrt {\frac{1+\frac{\sqrt 2}{2}}{2}}=\frac{\sqrt {2+\sqrt 2}}{2}$
Step 2. $sec(\frac{15\pi}{8})=\frac{1}{cos(\frac{15\pi}{8})}=\frac{2}{\sqrt {2+\sqrt 2}}=\frac{2\sqrt {2+\sqrt 2}}{2+\sqrt 2}=(2-\sqrt 2)\sqrt {2+\sqrt 2}$