A) 1/2
B) 2
C) 4
D) 8
Correct Answer: C
Solution :
[c]:\[\tan 9{}^\circ -\tan 27{}^\circ -\tan 63{}^\circ +\tan 81{}^\circ \] \[=\tan 9{}^\circ -\tan 27{}^\circ -\cot 27{}^\circ +\cot 9{}^\circ \] \[=\left( \tan 9{}^\circ +\text{cot}9{}^\circ \right)-\left( \tan 27{}^\circ +\cot 27{}^\circ \right)\] \[=\frac{2}{\sin {{18}^{o}}}-\frac{2}{\sin {{54}^{o}}}\] \[=2\left\{ \frac{\sin {{54}^{o}}-\sin {{18}^{o}}}{\sin {{18}^{o}}\sin {{54}^{o}}} \right\}=2.\frac{2.\cos {{36}^{o}}.\sin {{18}^{o}}}{\sin {{18}^{o}}\sin {{54}^{o}}}=4\]
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