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