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Double and Half Angle Formulas

$\begin{equation*} \sin(2 \theta) = 2 \cdot \sin(\theta) \cdot \cos(\theta) \end{equation*}$

$\begin{equation*} \cos(2 \theta) = \cos^2(\theta) - \sin^2(\theta) \end{equation*}$

$\begin{equation*} \cos(2 \theta) = 2 \cdot \cos^2(\theta) - 1 \end{equation*}$

$\begin{equation*} \cos(2 \theta) = 1 - 2 \cdot \sin^2(\theta) \end{equation*}$

$\begin{equation*} \tan(2 \theta) = \frac{2 \cdot \tan(\theta)}{1 - \tan^2(\theta)} \end{equation*}$

$\begin{equation*} \sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos(\theta)}{2}} \end{equation*}$

$\begin{equation*} \cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}} \end{equation*}$

$\begin{equation*} \tan\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}} \end{equation*}$

$\begin{equation*} \tan\left(\frac{\theta}{2}\right) = \frac{1 - \cos(\theta)}{\sin(\theta)} \end{equation*}$

$\begin{equation*} \tan\left(\frac{\theta}{2}\right) = \frac{\sin(\theta)}{1 + \cos(\theta)} \end{equation*}$

Example

Solve $\sin^2(\theta) = 2 \cdot \cos^2\left(\frac{\theta}{2}\right)$ for $(0, 2\pi)$ .

$\begin{equation*} \sin^2(\theta) = 2 \left( \pm \sqrt{\frac{1 + \cos(\theta)}{2}} \right)^2 \end{equation*}$

$\begin{equation*} \sin^2(\theta) = 2 \left( \frac{1 + \cos(\theta)}{2} \right) \end{equation*}$

$\begin{equation*} \sin^2(\theta) = 1 + \cos(\theta) \end{equation*}$

$\begin{equation*} 1 - \cos^2(\theta) = 1 + \cos(\theta) \end{equation*}$

$\begin{equation*} -1 \text{ } -1 \end{equation*}$

$\begin{equation*} -\cos^2(\theta) = \cos(\theta) \end{equation*}$

$\begin{equation*} -\cos^2(\theta) -\cos(\theta) = 0 \end{equation*}$

$\begin{equation*} \cos(\theta) \left( -\cos(\theta) - 1 \right) = 0 \end{equation*}$

$\begin{equation*} \cos(\theta) = 0 \end{equation*}$

$\begin{equation*} \theta = \frac{\pi}{2} \text{ or } \frac{3\pi}{2} \end{equation*}$

$\begin{equation*} -\cos(\theta) - 1 = 0 \end{equation*}$

$\begin{equation*} -\cos(\theta) = 1 \end{equation*}$

$\begin{equation*} \cos(\theta) = -1 \end{equation*}$

$\begin{equation*} \theta = \pi \end{equation*}$

$\begin{equation*} \boxed{\theta = \frac{\pi}{2} \text{, } \frac{3\pi}{2} \text{ or } \pi} \end{equation*}$