#+TITLE:Trigonometric identities
#+SETUPFILE: ../math_options.org
#+LATEX_HEADER: \usepackage{bm}
#+LATEX_HEADER: \usepackage{mathtools}

* Disclaimer
This site as of now just a technology demonstration and its claims
should not be taken as true (even though I myself am pretty confident
they are)

* Pythagorean identities
\begin{align*}
&\cos^2\left(x\right) + \sin^2\left(x\right) = 1\\
&\tan^2\left(x\right) + 1                    = \sec^2\left(x\right)\\
&1                    + \cot^2\left(x\right) = \csc^2\left(x\right)
\end{align*}
* Sum of angles
\begin{align*}
&\sin\left(a \pm b\right) = \sin\left(a\right)\cos\left(b\right) \pm \cos\left(a\right)\sin\left(b\right)\\
&\cos\left(a \pm b\right) = \cos\left(a\right)\cos\left(b\right) \mp \sin\left(a\right)\sin\left(b\right)\\
&\tan(a \pm b) = \frac{\tan(a) \pm \tan(b)}{1 \mp \tan(a) \tan(b)}
\end{align*}
* Multiple angles
\begin{align*}
&\sin(2\theta) = 2\sin(\theta)\cos(\theta)\\
&\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta
\end{align*}
* Half-angle formulae
\begin{align*}
&\sin \frac{\theta}{2} = \pm \sqrt{\frac{1-\cos\theta}{2}}\\
&\cos \frac{\theta}{2} = \pm \sqrt{\frac{1+\cos\theta}{2}}
\end{align*}
* Power-reduction formulae
\begin{align*}
&\sin^2\theta = \frac{1 - \cos(2\theta)}{2}\\
&\cos^2\theta = \frac{1 + \cos(2\theta)}{2}
\end{align*}
* Product-to-sum formulae
\begin{align*}
&2 \cos\theta \cos\varphi = \cos(\theta-\varphi) + \cos(\theta+\varphi)\\
&2 \sin\theta \sin\varphi = \cos(\theta-\varphi) - \cos(\theta+\varphi)\\
&2 \sin\theta \cos\varphi = \sin(\theta+\varphi) + \sin(\theta-\varphi)\\
&2\cos\theta\sin\varphi = \sin(\theta+\varphi) - \sin(\theta-\varphi)
\end{align*}
* Sum-to-product formulae
\begin{align*}
&\sin\theta \pm \sin\varphi = 2 \sin\left(\frac{\theta\pm\varphi}{2}\right) \cos\left(\frac{\theta\mp\varphi}{2}\right)\\
&\cos\theta + \cos\varphi = 2 \cos\left(\frac{\theta+\varphi}{2}\right) \cos\left(\frac{\theta-\varphi}{2}\right)\\
&\tan\theta \pm \tan\varphi = \frac{\sin(\theta\pm\varphi)}{\cos\theta \cos\varphi}\\
&\cos\theta - \cos\varphi = - 2 \sin\left(\frac{\theta+\varphi}{2}\right) \sin\left(\frac{\theta-\varphi}{2}\right)
\end{align*}