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