Prove that: tan 36° + tan 9° + tan 36° tan 9° = 1

Question

Prove that:

\[ \tan36^\circ+\tan9^\circ+\tan36^\circ\tan9^\circ=1 \]

Proof

\[ \tan(36^\circ+9^\circ) = \frac{\tan36^\circ+\tan9^\circ} {1-\tan36^\circ\tan9^\circ} \]

\[ \tan45^\circ = \frac{\tan36^\circ+\tan9^\circ} {1-\tan36^\circ\tan9^\circ} \]

\[ 1 = \frac{\tan36^\circ+\tan9^\circ} {1-\tan36^\circ\tan9^\circ} \]

\[ 1-\tan36^\circ\tan9^\circ = \tan36^\circ+\tan9^\circ \]

\[ \tan36^\circ+\tan9^\circ+\tan36^\circ\tan9^\circ =1 \]

Hence proved.