Воспользуемся формулой косинуса двойного угла:
\(\cos 2\alpha = 2{\cos ^2}\alpha -1.\)
\(\cos \alpha = -\dfrac{7}{8}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,2{\cos ^2}\dfrac{\alpha }{2}-1 = -\dfrac{7}{8}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\cos ^2}\dfrac{\alpha }{2} = \dfrac{1}{{16}}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\cos \dfrac{\alpha }{2} = \dfrac{1}{4},\,\,\,\,}\\{\cos \dfrac{\alpha }{2} = -\dfrac{1}{4}.}\end{array}} \right.\)
Так как \(\alpha \, \in \,\left( {\pi ;\dfrac{{3\pi }}{2}} \right)\), то \(\dfrac{\alpha }{2}\, \in \,\left( {\dfrac{\pi }{2};\dfrac{{3\pi }}{4}} \right)\) (II четверть), значит \(\cos \dfrac{\alpha }{2} < 0\), то есть \(\cos \dfrac{\alpha }{2} = -\dfrac{1}{4} = -0,25.\)
Ответ: \(-0,25.\)