The value of [ 9^- 27^- 63^+ 81^] is __________.

Question

The value of $$\tan 9^{\circ}-\tan 27^{\circ}-\tan 63^{\circ}+\tan 81^{\circ}$$ is \_\_\_\_\_\_\_\_\_\_.

Answer: 4

Accepted Answer

$\begin{gathered} 	an {9}^{\circ }-	an {27}^{\circ }-	an {63}^{\circ }+	an {81}^{\circ } \ =	an {9}^{\circ }+	an ({90}^{\circ }-{9}^{\circ })-	an {27}^{\circ }-	an ({90}^{\circ }-{27}^{\circ }) \ =	an {9}^{\circ }+\cot {9}^{\circ }-	an {27}^{\circ }-\cot {27}^{\circ } \ =\frac{\sin {9}^{\circ }}{\cos {9}^{\circ }}+\frac{\cos {9}^{\circ }}{\sin {9}^{\circ }}-(\frac{\sin {27}^{\circ }}{\cos {27}^{\circ }}+\frac{\cos {27}^{\circ }}{\sin {27}^{\circ }}) \ =\frac{{\sin }^{2}{9}^{\circ }+{\cos }^{2}{9}^{\circ }}{\sin {9}^{\circ }\cos {9}^{\circ }}-(\frac{{\sin }^{2}{27}^{\circ }+{\cos }^{2}{27}^{\circ }}{\cos {27}^{\circ }\sin {27}^{\circ }}) \ =\frac{2}{\sin {18}^{\circ }}-\frac{2}{\sin {54}^{\circ }} \ =\frac{2	imes 4}{\sqrt{5}-1}-\frac{2	imes 4}{\sqrt{5}+1} \ =8(\frac{\sqrt{5}+1-\sqrt{5}+1}{5-1})=2(2)=4 \end{gathered}$

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