The value of tan 9 ^ ^ o - tan 27 ^ ^ o - tan 63 ^ ^ o + tan 81 ^ ^...

The value of

tan{9}^{{}^{o}} - tan{27}^{{}^{o}} - tan{63}^{{}^{o}} + tan{81}^{{}^{o}}

is _____.

Correct answer

Step-by-step solution

Given,
$tan{9}^{{}^{o}} - tan{27}^{{}^{o}} - tan{63}^{{}^{o}} + tan{81}^{{}^{o}}$
$= \left(cot{81}^{°} + tan{81}^{{}^{o}}\right) - \left(tan{27}^{°} + cot{27}^{{}^{o}}\right)$
$= \left(tan{9}^{°} + cot{9}^{°}\right) - \left(tan{27}^{°} + cot{27}^{°}\right)$
$= \left(\dfrac{sin{9}^{°}}{cos{9}^{°}} + \dfrac{cos{9}^{°}}{sin{9}^{°}}\right) - \left(\dfrac{sin{27}^{°}}{cos{27}^{°}} + \dfrac{cos{27}^{°}}{sin{27}^{°}}\right)$
$= \dfrac{2}{sin{18}^{o}} - \dfrac{2}{sin{54}^{o}}$
$= \left(\dfrac{2 \times 4}{\sqrt{5} - 1} - \dfrac{2 \times 4}{\sqrt{5} + 1}\right)$
$= 8\left(\dfrac{1}{\sqrt{5} - 1} - \dfrac{1}{\sqrt{5} + 1}\right)$
$= 8\left(\dfrac{\sqrt{5} + 1 - \left(\sqrt{5} - 1\right)}{\left(\sqrt{5} - 1\right)\left(\sqrt{5} + 1\right)}\right)$
$= 8\left(\dfrac{2}{5 - 1}\right) = 4$

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