Trig3.11

From Exampleproblems

Prove that $\cot \theta+\cot (60+\theta)-\cot (60-\theta)=3\cot 3\theta\,$

LHS= $\cot \theta+\frac{\cot 60 \cot \theta-1}{\cot 60+\cot \theta}-\frac{\cot 60 \cot \theta-1}{\cot \theta-\cot 60}\,$

$\cot \theta+\frac{\frac{1}{\sqrt{3}}\cot \theta-1}{\frac{1}{\sqrt{3}}+\cot \theta}-\frac{\frac{1}{\sqrt{3}}\cot \theta-1}{\cot \theta-\frac{1}{\sqrt{3}}}\,$

$\cot \theta+\frac{\cot \theta-\sqrt{3}}{1+\sqrt{3}\cot \theta}-\frac{\cot \theta+\sqrt{3}}{\sqrt{3}\cot \theta-1}\,$

$\cot \theta+\frac{(\cot\theta-\sqrt{3})(\sqrt{3}\cot \theta-1)-(\cot \theta+\sqrt{3})(1+\sqrt{3}\cot \theta)}{(\sqrt{3}\cot \theta+1)(\sqrt{3}\cot \theta-1)}\,$