$\cot{3\theta} \,=\, \dfrac{3\cot{\theta}-\cot^3{\theta}}{1-3\cot^2{\theta}}$
$\dfrac{3\cot{\theta}-\cot^3{\theta}}{1-3\cot^2{\theta}} \,=\, \cot{3\theta}$
It is called cot triple angle identity and used in two different cases as a formula.
The co-tangent of triple angle identity is used to either expand or simplify the triple angle cot functions like $\cot{3A}$, $\cot{3x}$, $\cot{3\alpha}$ and etc. For example,
$(1) \,\,\,\,\,\,$ $\cot{3x} \,=\, \dfrac{3\cot{x}-\cot^3{x}}{1-3\cot^2{x}}$
$(2) \,\,\,\,\,\,$ $\cot{3A} \,=\, \dfrac{3\cot{A}-\cot^3{A}}{1-3\cot^2{A}}$
$(3) \,\,\,\,\,\,$ $\cot{3\alpha} \,=\, \dfrac{3\cot{\alpha}-\cot^3{\alpha}}{1-3\cot^2{\alpha}}$
Learn how to derive the rule of cot triple angle identity in trigonometry by geometry.
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