# Double angle formulas

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## Statement

The "double angle formulas" refer to the following two formulas

• [ilmath]\forall \varphi,\psi\in\mathbb{R}[\sin(\varphi\pm\psi)\eq\sin(\varphi)\cos(\psi)\pm\cos(\varphi)\sin(\psi)][/ilmath]
• [ilmath]\forall\varphi,\psi\in\mathbb{R}[\cos(\varphi\pm\psi)\eq\cos(\varphi)\cos(\psi)\mp\sin(\varphi)\sin(\psi)][/ilmath]

However sometimes it is taken to mean the following two special cases:

• [ilmath]\forall\varphi\in\mathbb{R}[\sin(2\varphi)\eq2\sin(\varphi)\cos(\varphi)][/ilmath] and
• [ilmath]\forall\varphi\in\mathbb{R}\big[\cos(2\varphi)\eq(\cos(\varphi))^2-(\sin(\varphi))^2\big][/ilmath]
• Noting that [ilmath](\sin(\theta))^2+(\cos(\theta))^2\eq 1[/ilmath] we see that [ilmath](\cos(\theta))^2\eq 1-(\sin(\theta))^2[/ilmath] and [ilmath](\sin(\theta))^2\eq 1-(\cos(\theta))^2[/ilmath], yielding:
1. [ilmath]\forall\varphi\in\mathbb{R}\big[\cos(2\varphi)\eq 1-2(\sin(\varphi))^2\big][/ilmath] and
2. [ilmath]\forall\varphi\in\mathbb{R}\big[\cos(2\varphi)\eq 2(\cos(\varphi))^2-1\big][/ilmath]
Either form is commonplace.