# To prove that tan &#x2061;<!-- ⁡ -->

To prove that $\mathrm{tan}70=\mathrm{tan}20+2\mathrm{tan}50$
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Darrell Valencia
$\begin{array}{rl}\mathrm{tan}70-\mathrm{tan}20& =\frac{\mathrm{sin}70}{\mathrm{cos}70}-\frac{\mathrm{sin}20}{\mathrm{cos}20}\\ & =\frac{\mathrm{sin}70\mathrm{cos}20-\mathrm{cos}70\mathrm{sin}20}{\mathrm{cos}20\mathrm{cos}70}\\ & =\frac{\mathrm{sin}\left(70-20\right)}{\mathrm{cos}20\mathrm{cos}70}\\ & =\frac{\mathrm{sin}50}{\mathrm{cos}20\mathrm{cos}70}\\ & =\frac{\mathrm{sin}50}{\mathrm{cos}20.\mathrm{sin}\left(20\right)}\\ & =\frac{2\mathrm{sin}50}{2\mathrm{sin}20.\mathrm{cos}20}\\ & =\frac{2\mathrm{sin}50}{\mathrm{sin}40}\\ & =\frac{2\mathrm{sin}50}{\mathrm{cos}\left(90-40\right)}\\ & =\frac{2\mathrm{sin}50}{\mathrm{cos}50}\\ & =2\mathrm{tan}50\end{array}$