Question

Show , without using tables or calculators, that
$tan{9}^{\circ }-tan{27}^{\circ }-tan{63}^{\circ }+tan{18}^{\circ }=4$

Answer 1

The given expression
= tan $9^{\circ }$ - tan ${27}^{\circ }$ - cot ${27}^{\circ }$ + cot $9^{\circ }$
= (tan $9^{\circ }$ + cot $9^{\circ }$) - (tan ${27}^{\circ }$ + cot ${27}^{\circ }$)
= $\frac{1}{sin{9}^{\circ }cos{9}^{\circ }}-1sin{27}^{\circ }cos{27}^{\circ }$
$=\frac{2}{sin{18}^{\circ }}-\frac{2}{sin{54}^{\circ }}=\frac{2}{sin{18}^{\circ }}-\frac{2}{cos{36}^{\circ }}$
by complementary rule
$=2\left[\frac{4}{\sqrt{\left(5\right)}-1}\frac{4}{\sqrt{\left(5\right)}+1}\right]=8.\frac{2}{4}=4$