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$\frac{{\sqrt 2 - \sin \alpha - \cos \alpha }}{{\sin \alpha - \cos \alpha }} = $
$\sec \left( {\frac{\alpha }{2} - \frac{\pi }{8}} \right)$
$\cos \left( {\frac{\pi }{8} - \frac{\alpha }{2}} \right)$
$\tan \left( {\frac{\alpha }{2} - \frac{\pi }{8}} \right)$
$\cot \left( {\frac{\alpha }{2} - \frac{\pi }{2}} \right)$
Solution
(c) $\frac{{\sqrt 2 – \sin \alpha – \cos \alpha }}{{\sin \alpha – \cos \alpha }}$
$= \frac{{\sqrt 2 – \sqrt 2 \left\{ {\frac{1}{{\sqrt 2 }}\sin \alpha + \frac{1}{{\sqrt 2 }}\cos \alpha } \right\}}}{{\sqrt 2 \left\{ {\frac{1}{{\sqrt 2 }}\sin \alpha – \frac{1}{{\sqrt 2 }}\cos \alpha } \right\}}}$
$=\frac{{\sqrt 2 – \sqrt 2 \cos \left( {\alpha – \frac{\pi }{4}} \right)}}{{\sqrt 2 \sin \left( {\alpha – \frac{\pi }{4}} \right)}}$
$= \frac{{\sqrt 2 \left\{ {\,1 – \cos \theta } \right\}}}{{\sqrt 2 \sin \theta }},$ where $\theta = \alpha – \frac{\pi }{4}$
$= \frac{{2{{\sin }^2}(\theta /2)}}{{2\sin (\theta /2)\cos (\theta /2)}} = \tan \frac{\theta }{2}$
$ = \tan \left( {\frac{\alpha }{2} – \frac{\pi }{8}} \right)$.
