Answer
$1-\sqrt 2$
Work Step by Step
Recall the trigonometric identity for $tan$:
$\tan(\dfrac{\theta}{2})=\dfrac{1-\cos (\theta)}{\sin (\theta)}$
Therefore,
$\tan (\ \dfrac{7 \pi}{8})=\dfrac{1-\cos ( \ \dfrac{7 \pi}{4})}{\sin ( \ \dfrac{7 \pi}{4})}\\=\dfrac{1-(\dfrac{\sqrt 2}{2})}{-\dfrac{\sqrt 2}{2}}\\=-\dfrac{2 -\sqrt 2}{\sqrt 2} \\=1-\sqrt 2$
