Trigonometric identity via complex exponential

Noting that

$$\text{Re}[z_1z_2] = \text{Re}[z_1]\text{Re}[z_2]-\text{Im}[z_1]\text{Im}[z_2],$$
how can
$$\cos(\alpha+\beta) = \text{Re}\left[e^{j(\alpha+\beta)}\right]$$
be expressed? Give the final answer in a simple form, without complex-valued functions.

Source.

and here is my attempt:

We know that

$$\cos(\alpha+\beta) = \cos(\alpha)\cos(\beta)-\sin(\alpha)\sin(\beta).$$
So taking the real part of the cosine function gives
$$\text{Re}\left[\cos(\alpha)\right]\text{Re}[\cos(\beta)]-\text{Im}[\sin(\alpha)]\text{Im}[\sin(\beta)].$$

Now we use Euler's formula to convert sin and cosine to complex exponentials

$$\begin{align*} \cos x &= \text{Re}\{e^{ix}\} = \frac{e^{ix}+e^{-ix}}{2}\\ \sin x &= \text{Im}\{e^{ix}\} = \frac{e^{ix}-e^{-ix}}{2i} \end{align*}$$
$$= \left[\frac{e^{j\alpha}+e^{-j\alpha}}{2}\right]\left[\frac{e^{j\beta}+e^{-j\beta}}{2}\right]-\left[\frac{e^{j\alpha}-e^{-j\alpha}}{2j}\right]\left[\frac{e^{j\beta}-e^{-j\beta}}{2j}\right]$$
Now once I do this, I will cancel out some terms, but my final answer is not $e^{\alpha+\beta}$ as it should be.

Source.

Please let me know what it is I am doing wrong.
Thanks

Answer

Hint:

You know that:

$$\cos (\alpha+\beta)=\mbox{Re}\left[e^{i(\alpha+\beta)}\right]=\mbox{Re}\left[e^{i\alpha}e^{i\beta}\right]$$
Now use the given rule for the real part of a product and you have done.

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From your rule you have:

$$\mbox{Re}\left[e^{i\alpha}e^{i\beta}\right]=\mbox{Re}\left[e^{i\alpha}\right]\mbox{Re}\left[e^{i\beta}\right]-\mbox{Im}\left[e^{i\alpha}\right]\mbox{Im}\left[e^{i\beta}\right]$$
and: $\mbox{Re}\left[e^{i\alpha}\right]=\cos \alpha$, $\mbox{Im}\left[e^{i\alpha}\right]=\sin \alpha$, $\mbox{Re}\left[e^{i\beta}\right]=\cos \beta$, $\mbox{Im}\left[e^{i\beta}\right]=\sin \beta$