tan10°+tan50°=tan60°-tan60°tan10°tan50° Why does this equation hold?

Tan10 ° + tan50 ° = sin (10 + 50) / (cos10 * cos50) = sin60 / (cos10 cos50) = radical 3 / (2cos10 cos50)
tan60°-tan60°tan10°tan50°
=tan60(1-tan10tan50)
=tan60(1-sin10sin50/cos10cos50)
=tan60(cos10cos50-sin10sin50)/cos10cos50
=tan60*cos(10+50)/cos10cos50
=(sin60/cos60)*cos60/cos10cos50
=sin60/cos10cos50
=Root 3 / (2cos10cos50)
So: tan10 ° + tan50 ° = tan60 ° - tan60 ° tan10 ° tan50 °

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