Prove that: tan(π/12) + tan(π/6) + tan(π/12)tan(π/6) = 1

Question

Prove that:

\[ \tan\frac{\pi}{12} + \tan\frac{\pi}{6} + \tan\frac{\pi}{12}\tan\frac{\pi}{6} =1 \]

Proof

\[ \tan\left(\frac{\pi}{12}+\frac{\pi}{6}\right) = \frac{ \tan\frac{\pi}{12}+\tan\frac{\pi}{6} }{ 1-\tan\frac{\pi}{12}\tan\frac{\pi}{6} } \]

\[ \tan\frac{\pi}{4} = \frac{ \tan\frac{\pi}{12}+\tan\frac{\pi}{6} }{ 1-\tan\frac{\pi}{12}\tan\frac{\pi}{6} } \]

\[ 1 = \frac{ \tan\frac{\pi}{12}+\tan\frac{\pi}{6} }{ 1-\tan\frac{\pi}{12}\tan\frac{\pi}{6} } \]

\[ 1-\tan\frac{\pi}{12}\tan\frac{\pi}{6} = \tan\frac{\pi}{12}+\tan\frac{\pi}{6} \]

\[ \tan\frac{\pi}{12} + \tan\frac{\pi}{6} + \tan\frac{\pi}{12}\tan\frac{\pi}{6} =1 \]

Hence proved.