Question
$\tan 9^\circ - \tan 27^\circ - \tan 63^\circ + \tan 81^\circ = $
$ = \tan \,\,{9^o} - \tan \,\,{27^o} - \cot \,\,{27^o} + \cot \,\,{9^o}$
$ = (\tan \,\,{9^o} + \cot \,\,{9^o}) - (\tan \,\,{27^o} + \cot \,\,{27^o})$
$ = \frac{{\cos ({9^o} - {9^o})}}{{\sin {9^o}\cos {9^o}}} - \frac{{\cos ({{27}^o} - {{27}^o})}}{{\sin {{27}^o}.\cos {{27}^o}}} = \frac{2}{{\sin {{18}^o}}} - \frac{2}{{\sin {{54}^o}}}$
$ = 2\,\left\{ {\frac{{\sin \,\,{{54}^o} - \sin \,\,{{18}^o}}}{{\sin \,\,{{18}^o}\,\sin \,\,{{54}^o}}}} \right\} $
$= 2.\,\,\frac{{2\,.\,\cos \,\,{{36}^o}.\,\sin \,\,{{18}^o}}}{{\sin \,\,{{18}^o}.\,\sin \,\,{{54}^o}}} = 4$
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