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Can't use algebra, you say? Nonsense! All you need is [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula#Relationship_to_trigonometry)
$$
e^{i\theta} = \cos\theta + i\sin\theta
$$
and these identities just fall out of the algebra. For example:

\begin{align}
\sin (\alpha + \beta) &= \frac1{2i} \left(e^{i(\alpha + \beta)} - e^{-i(\alpha + \beta)}\right)\\\\
&= \frac1{2i} \left(e^{i\alpha}e^{i\beta} - e^{-i\alpha}e^{-i\beta}\right)\\\\
&= \frac1{2i} \left((\cos\alpha + i\sin\alpha)(\cos\beta + i\sin\beta) - (\cos\alpha - i\sin\alpha)(\cos\beta - i\sin\beta)\right)\\\\
&= \frac1{2i} \left(\cos\alpha\cos\beta + i\sin\alpha\cos\beta + i\cos\alpha\sin\beta - \sin\alpha\sin\beta\right.\\\\
&\left.\qquad\qquad- \cos\alpha\cos\beta + i\sin\alpha\cos\beta + i\cos\alpha\sin\beta + \sin\alpha\sin\beta\right)\\\\
&= \frac1{2i} \left(2i\sin\alpha\cos\beta + 2i\cos\alpha\sin\beta\right)\\\\
&= \sin\alpha\cos\beta + \cos\alpha\sin\beta
\end{align}