\begin{equation*}
\sin\left(\theta\right)
=
\tan\left(\theta\right)
\cos\left(\theta\right)
\quad
\text{ 🙵 }
\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{ 🙵 }
\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{ 🙵 }
\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*}