Tangents Of Multiples Of Nine

$\tan 81^{\circ} - \tan 63^{\circ} - \tan 27^{\circ} + \tan 9^{\circ}\\ =\tan 81^{\circ} - \tan 63^{\circ} - \color{#3D99F6}{\cot 63^{\circ}} + \color{#3D99F6}{\cot 81^{\circ}}\\ =\tan 81^{\circ} + \color{#EC7300}{\dfrac{1}{\tan 81^{\circ}}} - \tan 63^{\circ} - \color{#EC7300}{\dfrac{1}{\tan 63^{\circ}}}\\ =\dfrac{\tan^2 81^{\circ} + 1}{\tan 81^{\circ}} - \dfrac{\tan^2 63^{\circ}+1}{\tan 63^{\circ}}\\ =\dfrac{\color{magenta}{\sec^2 81^{\circ}}}{\tan 81^{\circ}} - \dfrac{\color{magenta}{\sec^2 63^{\circ}}}{\tan 63^{\circ}}\\ =\dfrac{\color{teal}{\frac{1}{\cos^2 81^{\circ}}}}{\color{#BBBBBB}{\frac{\sin 81^{\circ}}{\cos 81^{\circ}}}} - \dfrac{\color{teal}{\frac{1}{\cos^2 63^{\circ}}}}{\color{#BBBBBB}{\frac{\sin 63^{\circ}}{\cos 63^{\circ}}}}\\ =\dfrac{\cos 81^{\circ}}{\cos^2 81^{\circ} \sin 81^{\circ}} - \dfrac{\cos 63^{\circ}}{\cos^2 63^{\circ}\sin 63^{\circ}}\\ =\dfrac{1}{\cos 81^{\circ}\sin 81^{\circ}} - \dfrac{1}{\cos 63^{\circ}\sin 63^{\circ}}\\ =\dfrac{2}{2\cos 81^{\circ}\sin 81^{\circ}} - \dfrac{2}{2\cos 63^{\circ}\sin 63^{\circ}}\\ =\dfrac{2}{\color{#69047E}{\sin 162^{\circ}}} - \dfrac{2}{\color{#69047E}{\sin 126^{\circ}}}\\ =\dfrac{2}{\color{#20A900}{\sin 18^{\circ}}} - \dfrac{2}{\color{#20A900}{\sin 54^{\circ}}}\\ =2\left(\dfrac{\sin 54^{\circ} - \sin 18^{\circ}}{\sin 54^{\circ}\sin 18^{\circ}}\right)\\ =4\left(\dfrac{\sin 54^{\circ}-\sin 18^{\circ}}{2\sin 54^{\circ}\sin 18^{\circ}}\right)\\ =4\left(\dfrac{\sin 54^{\circ}-\sin 18^{\circ}}{\color{#27D2E7}{\cos 36^{\circ} - \cos 72^{\circ}}}\right)\\ =4\left(\dfrac{\sin 54^{\circ} - \sin 18^{\circ}}{\color{#D61F06}{\sin 54^{\circ}} - \color{#D61F06}{\sin 18^{\circ}}}\right)\\ =\boxed{4}$

---

Formulas used:

1. Complementary angles - For $0^{\circ}<x<90^{\circ},\;\color{#3D99F6}{\tan x = \cot(90^{\circ}-x)}$
2. Definition of cotangent - $\color{#EC7300}{\cot x = \dfrac{1}{\tan x}}$
3. Pythagorean trigonometric identities - $\color{magenta}{\tan ^2 x + 1 = \sec^2 x}$
4. Definition of secant - $\sec x = \dfrac{1}{\cos x} \implies \color{teal}{\sec^2 x = \dfrac{1}{\cos^2 x}}$
5. Definition of tangent - $\color{#BBBBBB}{\tan x = \dfrac{\sin x}{\cos x}}$
6. Double angle formula - $\color{#69047E}{2\sin x \cos x = \sin 2x}$
7. Unit circle and reference angles - For $90^{\circ} < x < 180^{\circ},\;\color{#20A900}{\sin x = \sin(180^{\circ} - x)}$
8. Product to sum formula - $\sin a \sin b = \dfrac{1}{2} \Big(\cos(a-b) - \cos(a+b) \Big)\implies \color{#27D2E7}{2 \sin a \sin b = \cos(a-b) - \cos(a+b)}$
9. Complementary angles - For $0^{\circ}<x<90^{\circ},\;\color{#D61F06}{\cos x = \sin(90^{\circ}-x)}$

Relevant wiki: Sum and Difference Trigonometric Formulas - Problem Solving

$(\tan 81^\circ+\tan9^\circ)-(\tan 63^\circ+\tan27^\circ)$

$=\left(\dfrac{\overbrace{\sin 90^{\circ}}^{\color{#D61F06}{1}}}{\underbrace{\cos 9^{\circ}\underbrace{\cos 81^{\circ}}_{\sin9^{\circ}}}_{\color{#D61F06}{\dfrac{\sin 18^{\circ}}{2}}}}\right)-\left( \dfrac{\overbrace{\sin 90^{\circ}}^{\color{#D61F06}{1}}}{\underbrace{\cos 27^{\circ}\underbrace{\cos 63^{\circ}}_{\sin 27^{\circ}}}_{\color{#D61F06}{\dfrac{\sin 54^{\circ}}2}}}\right)$

$\large=2\left(\dfrac{1}{\frac{\sqrt 5-1}{4}}-\dfrac{1}{\frac{\sqrt 5+1}{4}}\right)$

$\huge =2\times 2=\boxed{\color{#3D99F6}{4}}$

---

In second line I used : $\small{\color{teal}{\tan A+\tan B=\dfrac{\sin (A+B)}{\cos A\cos B}}}$

Also, $\small{\sin 18^{\circ}=\dfrac{\sqrt 5-1}{4},\sin 54^{\circ}=\dfrac{\sqrt 5+1}{4}}$

$\begin{aligned} X & = \tan 81^\circ - \tan 63^\circ - \tan 27^\circ + \tan 9^\circ \\ & = (\tan 81^\circ + \tan 9^\circ) - (\tan 63^\circ + \tan 27^\circ) \\ & = \left(\frac {\sin 81^\circ}{\cos 81^\circ} + \frac {\sin 9^\circ}{\cos 9^\circ} \right) - \left(\frac {\sin 63^\circ}{\cos 63^\circ} + \frac {\sin 27^\circ}{\cos 27^\circ} \right) \\ & = \frac {\sin 81^\circ \cos 9^\circ + \cos 81^\circ \sin 9^\circ}{\sin 9^\circ \cos 9^\circ} - \frac {\sin 63^\circ \cos 27^\circ + \cos 63^\circ \sin 27^\circ}{\sin 27^\circ \cos 27^\circ} \\ & = \frac {\sin 90^\circ}{\frac 12 \sin 18^\circ} - \frac {\sin 90^\circ}{\frac 12 \sin 54^\circ} \\ & = \frac 2{\sin 18^\circ} - \frac 2{\sin 54^\circ} \\ & = \frac 2{\cos 72^\circ} - \frac 2{\cos 36^\circ} \\ & = \frac {2(\cos 36^\circ - \cos 72^\circ)}{\cos 36^\circ\cos 72^\circ} \\ & = \frac {2(\color{#3D99F6}{\cos 36^\circ + \cos 108^\circ})}{\color{#D61F06}{\cos 36^\circ\cos 72^\circ}} \quad \quad \small \color{#3D99F6}{\text{Note that }\cos \frac \pi 5 + \cos \frac {3\pi}5 = \frac 12} \\ & = \frac {2\cdot \color{#3D99F6}{\frac 12}}{\color{#D61F06}{\frac 14}} = \boxed{4} \quad \quad \small \color{#D61F06}{\frac {\sin 36^\circ\cos 36^\circ\cos 72^\circ}{\sin 36^\circ} = \frac {\sin 72^\circ \cos 72^\circ}{2\sin 36^\circ} = \frac {\sin 144^\circ}{4 \sin 36^\circ} = \frac {\sin 36^\circ}{4 \sin 36^\circ} = \frac 14} \end{aligned}$