The value of tan 9° - tan 27° - tan 63° + tan 81⁰, is

Solution:

We have,

$\begin{array}{l}\tan 9^{\circ }-\tan {27}^{\circ }-\tan {63}^{\circ }+\tan {81}^{\circ }\\ =\left(\tan 9^{\circ }+\tan {81}^{\circ }\right)-\left(\tan {27}^{\circ }+\tan {63}^{\circ }\right)\\ =\frac{1}{\cos 9^{\circ }\cos {81}^{\circ }}-\frac{1}{\cos {27}^{\circ }\cos {63}^{\circ }}\\ =\frac{1}{\sin 9^{\circ }\cos 9^{\circ }}-\frac{1}{\sin {27}^{\circ }\cos {27}^{\circ }}\\ =\frac{2}{\sin {18}^{\circ }}-\frac{2}{\sin {54}^{\circ }}\\ =2\left\{\frac{\sin 54-\sin {18}^{\circ }}{\sin {54}^{\circ }\sin {18}^{\circ }}\right\}=2\left\{\frac{2\cos {36}^{\circ }\sin {18}^{\circ }}{\sin {18}^{\circ }\cos {36}^{\circ }}\right\}=4\end{array}$