${\cos }^{-1}\left\{\frac{1}{\sqrt{2}}(\cos \frac{9\pi }{10}-\sin \frac{9\pi }{10})\right\}=$

${\cos }^{-1}\left[\cos \left(\frac{9\pi }{10}+\frac{\pi }{4}\right)\right]={\cos }^{-1}\left[\cos \left(\frac{23\pi }{20}\right)\right]$

                                                    $\begin{array}{l}={\cos }^{-1}\left[\cos \left(2\pi -\frac{17\pi }{20}\right)\right]\\ ={\cos }^{-1}\left[\cos \left(\frac{17\pi }{20}\right)\right]=\frac{17\pi }{20}\end{array}$