\begin{equation*} \sin\left(\theta\right) = \tan\left(\theta\right) \cos\left(\theta\right) \quad \text{ ðŸ™µ } \quad \csc\left(\theta\right) = \cot\left(\theta\right) \sec\left(\theta\right) \end{equation*} \begin{equation*} \sin\left(\theta\right)\csc\left(\theta\right) \;=\; \cos\left(\theta\right)\sec\left(\theta\right) \;=\; \tan\left(\theta\right)\cot\left(\theta\right) \;=\; 1 \end{equation*} \begin{equation*} \sin^2\left(\theta\right) + \cos^2\left(\theta\right) \;=\; \sec^2\left(\theta\right) - \tan^2\left(\theta\right) \;=\; \csc^2\left(\theta\right) - \cot^2\left(\theta\right) \;=\; 1 \end{equation*} \begin{equation*} \sin\left(2\theta\right) = 2\sin\left(\theta\right)\cos\left(\theta\right) \quad \text{ ðŸ™µ } \quad \cos\left(2\theta\right) = \cos^2\left(\theta\right) - \sin^2\left(\theta\right) \end{equation*} \begin{equation*} 2\sin^2\left({\textstyle\frac{1}{2}}\theta\right) = 1-\cos\left(\theta\right) \quad \text{ ðŸ™µ } \quad 2\cos^2\left({\textstyle\frac{1}{2}}\theta\right) = 1+\cos\left(\theta\right) \end{equation*}
\begin{equation*} \mathrm{e}^{x} = y \;\iff\; \ln(y) = x \end{equation*}
\begin{equation*} \int\tan(x)\,\mathrm{d}x = \ln|\sec(x)| +C \qquad \int\ln(x)\,\mathrm{d}x = x\ln|x| - x +C \end{equation*} \begin{equation*} \int\sec(x)\,\mathrm{d}x = \ln\left|\sec(x) + \tan(x)\right| + C \,. \end{equation*}