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Pythagorean Identities

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

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

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

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$\begin{equation*} a^2 + b^2 = c^2 \end{equation*}$

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

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

Example

If $\tan(\theta) = -7$ and $\sin(\theta) > 0$ , find $\cos(\theta)$ .

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

$\begin{equation*} (-7)^2 + 1 = \sec^2 (\theta) \end{equation*}$

$\begin{equation*} 49 + 1 = \sec^2 (\theta) \end{equation*}$

$\begin{equation*} \sqrt{50} = \sqrt{\sec^2 (\theta)} \end{equation*}$

$\begin{equation*} \pm \sqrt{50} = \sec (\theta) \end{equation*}$

$\begin{equation*} \pm \sqrt{50} = \frac{1}{\cos(\theta)} \end{equation*}$

$\begin{equation*} \pm \frac{1}{\sqrt{50}} = \frac{\cos(\theta)}{1} \end{equation*}$

$\begin{equation*} \boxed{\pm \frac{\sqrt{50}}{50} = \cos(\theta)} \end{equation*}$