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The expression given is:

\[ \sqrt{2} - 2\sqrt{2} \sin^2\left(\frac{15\pi}{8}\right). \]

This can be simplified using the identity:

\[ 1 - 2\sin^2(x) = \cos(2x). \]

Therefore:

\[ \sqrt{2} - 2\sqrt{2} \sin^2\left(\frac{15\pi}{8}\right) = \sqrt{2} \cos\left(\frac{15\pi}{4}\right). \]

Since \(\frac{15\pi}{4}\) is equivalent to adding \(7\pi/4\) to \(\pi\), this simplifies to:

\[ \sqrt{2} \cos\left(\pi + \frac{7\pi}{4}\right) = \sqrt{2} \cos\left(\frac{7\pi}{4}\right). \]

The cosine of \(\frac{7\pi}{4}\) is \(\frac{\sqrt{2}}{2}\), so the expression simplifies to:

\[ \sqrt{2} \times \frac{\sqrt{2}}{2} = 1. \]