Question

$\: Tan20° + Tan40° + \sqrt {3 } Tan20° Tan40°$ ​

Answer 1

$\underline{\underline{\mathfrak{\Large{Solution : }}}} \\ \\ \\ \sf\: Tan20\degree + Tan40\degree + \sqrt {3}\: Tan20\degree Tan40\degree \\ \\ \\ \sf\: Tan (20\degree + 40\degree) = Tan60\degree \\ \\ \\ \sf\dfrac {Tan20\degree + Tan40\degree}{1 - Tan20\degree \: Tan40\degree } = \sqrt {3} \\ \\ \\ \sf\: Tan20\degree + Tan40\degree = \sqrt {3}\:(1 - Tan20\degree \: Tan40\degree) \\ \\ \\ \sf\: Tan20\degree + Tan40\degree = \sqrt {3} - \sqrt {3} \: Tan20\degree \: Tan40\degree \\ \\ \\ \bf\: Tan20\degree + Tan40\degree + \sqrt {3} \: Tan20\degree \: Tan40\degree = \sqrt {3}$