Answer
$\dfrac{2}{\sqrt{{2 -\sqrt 2}}}$
Work Step by Step
The inverse Identity for secant can be expressed as:
$\csc a =\dfrac {1}{\sin a}$ or, $\csc a =(\sin a)^{-1}$
Therefore,
$\csc ({\dfrac{7 \pi }{8}})=( \sqrt{\dfrac{1 - \cos ({\dfrac{ 7 \pi }{4}})}{2}} )^{-1} \\=[ \sqrt{\dfrac{1 -\dfrac{\sqrt 2}{2}}{2}}]^{-1} \\=[\dfrac{\sqrt {2 -\sqrt 2}}{2}]^{-1}\\ = \dfrac{2}{\sqrt{{2 -\sqrt 2}}}$
