Okay, here we have this:
Considering the provided sin, we are going to calculate it, so we obtain the following:
[tex]\begin{gathered} \sin \mleft(\frac{7\pi}{8}\mright) \\ =\cos \mleft(\frac{\pi}{2}-\frac{7\pi}{8}\mright) \\ =\cos \mleft(-\frac{3\pi}{8}\mright) \\ =\cos \mleft(\frac{3\pi}{8}\mright) \\ =\cos \mleft(\frac{\frac{3\pi}{4}}{2}\mright) \\ =\sqrt{\frac{1+\cos\left(\frac{3\pi}{4}\right)}{2}} \\ =\sqrt{\frac{1+\left(-\frac{\sqrt{2}}{2}\right)}{2}} \\ =\sqrt{\frac{2-\sqrt{2}}{4}} \\ =\frac{\sqrt{2-\sqrt{2}}}{2} \end{gathered}[/tex]
Finally we obtain that the correct answer is the option D.
