There are three identities which are essential -- all the others can be derived from these three:

• The Pythagorean Theorem: \begin{equation*}\sin^2(x)+\cos^2(x)=1\end{equation*}
• The cosine of a sum: \begin{equation*}\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)\end{equation*}
• The sine of a sum: \begin{equation*}\sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)\end{equation*}

I always tell my students that they ought to be able to graph both sine and cosine on an interval of length \(2\pi\) (the period of each), and do it within a minute: Sage does it more quickly...:)