Validation crisis in mathematical physics

Not sure if the problem is that I'm not a mathematician, but proofs are too short to be understandable to me. As an example,

De Moivre's Formula
(cos x + isin x)n = cos(nx) + isin(nx)

Proof:
(1) (eix)n = einx
(2) (cos x + i sin x)n = cos(nx) + isin(nx)  [Applying Euler's formula above]

source: https://fermatslasttheorem.blogspot.com/2006/02/eulers-formula.html

Happily, Wikipedia provides a more detailed proof by induction with more steps.

The truth of de Moivre's theorem can be established by using mathematical induction for natural numbers, and extended to all integers from there. For an integer $n$ , call the following statement $S (n)$ :

\[(\cos x + i \sin x)^n = \cos nx + i \sin nx.\]

For $n > 0$ , we proceed by mathematical induction. $S (1)$ is clearly true. For our hypothesis, we assume $S (k)$ is true for some natural $k$ . That is, we assume

\[\left(\cos x + i \sin x\right)^k = \cos kx + i \sin kx. \]

Now, considering $S (k + 1)$ :

\begin{alignat}{2} \left(\cos x+i\sin x\right)^{k+1} & = \left(\cos x+i\sin x\right)^{k} \left(\cos x+i\sin x\right)\\ & = \left(\cos kx + i\sin kx \right) \left(\cos x+i\sin x\right) &&\qquad \text{via induction hypothesis}\\ & = \cos kx \cos x - \sin kx \sin x + i \left(\cos kx \sin x + \sin kx \cos x\right)\\ & = \cos ((k+1)x) + i\sin ((k+1)x) &&\qquad \text{via trigonometric identities} \end{alignat}

See angle sum and difference identities.

We deduce that $S (k)$ implies $S (k + 1)$ . By the principle of mathematical induction it follows that the result is true for all natural numbers. Now, $S (0)$ is clearly true since $cos(0 x) + i sin(0 x) = 1 + 0 i = 1$ . Finally, for the negative integer cases, we consider an exponent of $- n$ for natural $n$ .

\begin{align} \left(\cos x + i\sin x\right)^{-n} & = \big( \left(\cos x + i\sin x\right)^n \big)^{-1} \\ & = \left(\cos nx + i\sin nx\right)^{-1} \\ & = \cos nx - i\sin nx \qquad\qquad(*)\\ & = \cos(-nx) + i\sin (-nx).\\ \end{align}

The equation (*) is a result of the identity

\[z^{-1} = \frac{\bar z}{|z|^2},\]

for $z = cos nx + i sin nx$ . Hence, $S (n)$ holds for all integers $n$ .