$\cos{3\theta} \,=\, 4\cos^3{\theta}-3\cos{\theta}$
$4\cos^3{\theta}-3\cos{\theta} \,=\, \cos{3\theta}$
It is called cos triple angle identity and used as a formula in two various cases.
Cosine of triple angle identity is used to either expand or simplify the triple angle cos functions like $\cos{3x}$, $\cos{3A}$, $\cos{3\alpha}$ and etc. For example,
$(1) \,\,\,\,\,\,$ $\cos{3x} \,=\, 4\cos^3{x}-3\cos{x}$
$(2) \,\,\,\,\,\,$ $\cos{3A} \,=\, 4\cos^3{A}-3\cos{A}$
$(3) \,\,\,\,\,\,$ $\cos{3\alpha} \,=\, 4\cos^3{\alpha}-3\cos{\alpha}$
Learn how to derive the rule of cos triple angle identity by geometric approach in trigonometry.
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