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...:)

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f(x) = sin(x)

g(x) = cos(x)

p1=plot(f, (x, -pi, pi), color='green', thickness=3)

p2=plot(g, (x, -pi, pi), color='blue', thickness=3)

p1+p2