<s>[INST] How to remember the trigonometric identities I have a test tomorrow and I am having trouble remembering those pesky trigonometrical identities (such as $1-\cos x=2\sin^2(\frac{x}{2})$ )
Do you guys have any tips on how I can remember these?
Thanks :)
 [/INST] My favourite trick: I don't remember any of them.  :-)  The only thing I have in mind is that this matrix
$$
\begin{pmatrix}
\cos\theta  & -\sin\theta \\
\sin\theta  & \cos\theta
\end{pmatrix}
$$
rotates vectors in the plane by an angle $\theta$ and matrix multiplication is the same as composition. Hence, you have identities like
$$
\begin{pmatrix}
\cos(2\theta)  & -\sin(2\theta) \\
\sin(2\theta)  & \cos(2\theta)
\end{pmatrix}

=

\begin{pmatrix}
\cos\theta  & -\sin\theta \\
\sin\theta  & \cos\theta
\end{pmatrix}

\begin{pmatrix}
\cos\theta  & -\sin\theta \\
\sin\theta  & \cos\theta
\end{pmatrix}
$$
from which it follows
$$
\cos(2\theta) = \cos^2\theta - \sin^2\theta
$$
and
$$
\sin(2\theta) = 2\sin\theta\cos\theta \ .
$$
Alternatively, as yoyo says, you could use Euler's identity,
$$
e^{i\theta} = \cos\theta + i \sin\theta
$$
to find, for instance, that
$$
\cos(\theta + \phi) + i\sin(\theta + \phi) = e^{i(\theta + \phi)} = e^{i\theta}e^{i\phi} = (\cos\theta + i\sin\theta) (\cos\phi + i\sin\phi)   \ .
$$
Hence,
$$
\cos(\theta + \phi) = \cos\theta\cos\phi - \sin\theta\sin\phi
$$
and
$$
\sin(\theta + \phi) = \sin\theta\cos\phi  + \cos\theta\sin\phi \ .
$$
</s>