The proof: https://youtu.be/p6j2nZKwf20
For context, here's the main idea of the proof. Using the definition of sine, we have $c^2 = \dfrac{2ab}{\sin 2\alpha}.$ Our goal is to show that the RHS is equivalent to $a^2 + b^2.$ If $a=b,$ then this is obvious since $\alpha = 45^\circ.$ Otherwise, if $a \neq b,$ then we perform a geometric construction called the "waffle cone" and do some hairy algebra to get to the desired result (this forms the bulk of the proof).
- How can you apply above mentioned proof for right angled isosceles triangles?
The proof is for general right triangles. An isosceles right triangle is a particular case of a general right triangle, so the proof already applies to isosceles right triangles (let $b=a$).
While it's true that the "waffle cone" construction requires $a<b,$ you don't need to do the construction if $a=b.$ The whole point of the construction is to obtain $$a^2 + b^2 = \dfrac{2ab}{\sin 2\alpha},$$ but if $a=b$ then $\alpha = 45^\circ$ and the above reduces to $$a^2 + a^2 = \dfrac{2 \cdot a \cdot a}{\sin 90^\circ}$$ which is true.
- Can you say that approach is trigonometrical because of using trigonometrical formula $\sin(2 \theta) = 2\sin(\theta)\cos(\theta)$ since it also can be derived geometrically by right angled triangles?
When they call it a trigonometric proof, all they're saying is that it uses trigonometric functions. That's it.
Sure, you could write the proof without using trigonometric functions. You could reduce everything down to simple algebra on similar triangles, explicitly writing $b/c$ in place of $\sin \beta.$
But does that mean it's not trigonometric? No. You could apply the same reasoning to argue ridiculous things, e.g. that the Fundamental Theorem of Calculus
$$\int_a^b F'(x) \, \textrm dx = F(b) - F(a)$$
is not actually differential/integral calculus because you can reduce it down to just limits:
$$\lim_{n \to 0} \sum_{i=0}^{n-1} \left( \lim_{h \to 0} \frac{F(x_i +h)-F(x_i)}{h} \right) \Delta x = F(b) - F(a)$$
where $\Delta x = \dfrac{b-a}{n}$ and $x_i = a + i \Delta x.$
So, when we say the proof is trigonometric, we mean that it uses trigonometric notation to describe symbolic patterns leveraged in trigonometry, not that it is impossible to state without trigonometric notation.
- Is it not possible to complete the proof by just adding the squares of two perpendicular sides without using the formula $\sin(2 \theta) = 2\sin(\theta)\cos(\theta)$ and say that proof is geometrical?
Here my question is related to how we have to update students using new developments in mathematics, therefore I hope you will not misunderstand this as a mathematics question.
This question is impossible to answer because the double-angle formula $\sin(2 \theta) = 2\sin(\theta)\cos(\theta)$ is not even used in the proof.
The only trigonometry that's used is the definition of sine and the law of sines. And the law of sines does not rely on the double-angle formula -- it follows directly from the definition of sine.
