A geometry problem by Elan Fenatic

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

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

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

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

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In second line I used : $\small{\color{teal}{\tan A+\tan B=\dfrac{\sin (A+B)}{\cos A\cos B}~~,\sin A\cos A=\dfrac{\sin 2A}2}}$

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

TanA + CotA=2Cosec2A