$\cos{2\theta} \,=\, \cos^2{\theta}-\sin^2{\theta}$

$\cos^2{\theta}-\sin^2{\theta} \,=\, \cos{2\theta}$

It is called cos double angle identity and used as a formula in two cases.

- Cos of double angle is expanded as the subtraction of squares of sin function from cos function.
- The subtraction of squares of sin function from cos function is simplified as cos of double angle.

Cosine of double angle identity is used to either expand or simplify the double angle functions like $\cos{2x}$, $\cos{2A}$, $\cos{2\alpha}$ and etc. For example,

$(1) \,\,\,\,\,\,$ $\cos{2x} \,=\, \cos^2{x}-\sin^2{x}$

$(2) \,\,\,\,\,\,$ $\cos{2A} \,=\, \cos^2{A}-\sin^2{A}$

$(3) \,\,\,\,\,\,$ $\cos{2\alpha} \,=\, \cos^2{\alpha}-\sin^2{\alpha}$

Learn how to derive the rule of cos double angle identity by geometric method in trigonometry.

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