The Principal Value Of Cos-1{1/Sqrt2 (Cos9Pi/10 - Sin9Pi/10)} Is

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

=\( cos^{-1}\left [cos 45 cos(\frac{9\pi }{10}) – sin 45 sin (\frac{9\pi }{10}) \right] \)

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

=\( cos^{-1}\left [cos (\frac{23\pi}{20}) \right] \)

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

=\( cos^{-1}\left [cos (\frac{3\pi}{20}) \right] \)

= \(cos^{-1}\left [-cos (\frac{3\pi}{20}) \right] \)

=\( \pi – cos^{-1}\left [cos (\frac{3\pi}{20}) \right] \)

=\( \pi – \frac{3\pi}{20} \)

= \(\frac{17\pi}{20} \)

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