For (1), we start with 2 definitions of cos(x) and sin(x), using their Taylor expansions

cos(x) = 1 - x^2^/2! + x^4^/4! - …

sin(x) = x - x^3^/3! + x^5^/5! - …

We now start with the definition of e^x^ Taylor expansion, and proceed to do some substitutions:

e^x^ = 1 + x + x^2^/2! + x^3^/3! + … + x^n^/n!

We can then substitute in: x=iθ (remembering that i^2^ = -1) to get

e^iθ^ = 1 + iθ - θ^2^/2! - iθ^3^/3! + θ^4^/4! + iθ^5^/5! + … etc…

If we group by real and complex, we can arrange the above as:

e^iθ^ = (1 - θ^2^/2! + θ^4^/4! + … ) + i(θ - θ^3^/3! + θ^5^/5! + … )

You should now realise that the left part resembles the expansion of cos(θ), and the right part resembles sin(θ). That is:

e^iθ^ = cos(θ) + i sin(θ)

Finally, we substitute in θ = π

e^iπ^ = cos(π) + i sin(π)

And we know that cos(π) = -1, and that sin(π) = 0, meaning that we end up with

e^iπ^ = -1 + i 0

or

e^iπ^ + 1 = 0

The teacher got excited because it is literally one of the most beautiful mathematical statements you can get, that connects five universal identities under a single statement: 0, 1, e, i, and π – and does so using 3 different operators (times, power, plus).

For (2), I’m still waiting as I think it’s currently holding the world together by sheer mass alone