I was wondering if quantum particles do actually exists in two different states simultaneously and if it has been proven they do indeed exists in a superposition of states.
How has it been figured out since observing it would collapse the wave-function into one single state (two superposition of states into one) as it has been mentioned in the Schrodinger's cat experiment?
I was wondering if quantum particles do actually exists in two different states simultaneously and if it has been proven they do indeed exists in a superposition of states
This is a common misconception. "Superposition of states" does not mean "existing in multiple states". Quantum systems are only in one state at a time, and this state can be expressed as a superposition of other, basis states. An analogy is how a vector can be expressed as a superposition of unit vectors (components), but it's still a single vector.
We mathematically express quantum states as superpositions because they tell us the probability of measuring a certain outcome. For example, if we have energy states $|E_n\rangle$, then we can express our quantum state $|\psi\rangle$ as $$|\psi\rangle=\sum_na_n|E_n\rangle$$ where the probability of measuring the energy $E_n$ is equal to $|a_n|^2$ when $\sum_n|a_n|^2=1$. This is one of the key features of quantum mechanics.
Since this mathematical formalism and its physical interpretation give rise to correct descriptions of the universe, I would say that yes, we have proven that quantum states exist in superpositions. However, at the end of the day superposition is just a mathematical idea; it's not physical.
how has it been figured out since observing it would collapse the wave-function into one single state (two superposition of states into one ) as it has been mentioned in the Schrodinger's cat experiment?
Description of wave function collapse depends on how you view quantum mechanics, and things are moving more towards "decoherence" rather than "wave function collapse", but for a (one of many?) coarse grained explanation, collapse is just how we describe what happens to our understanding of the wave function. When we make a measurement we change the system and what we know about it, so we have to "update" the wave function to reflect what we now know (and what we now "don't know" as well).
Considering all of the above then, wave function collapse does not prevent us from observing superpositions. Collapse and superpositions are all mathematical. It's not like we have a device that can "detect wave functions". However, superposition and collapse have physical implications according to how the quantum mechanics relates its formalism to the physical universe. Since these implications hold true (when valid), we can say these descriptions are valid in the context of the theory.
We need to make a careful distinction here between what is happening mathematically and what we are justified in interpreting physically.
It is true that in quantum mechanics the state of a system at any given moment is represented mathematically by a vector$^1$, and that vector may be written as a linear combination of basis vectors, each of which may physically represent a state that we may find the system in after we make a measurement.
A classic example is the two-state quantum system that represents the spin-state of a spin-1/2 quantum particle. We can in general write the state of the system as a linear combination of "up" and "down" spin states: $$|\psi\rangle=\alpha|\uparrow\rangle+\beta|\downarrow\rangle \tag{1}. $$ It is correct to say that both $|\uparrow\rangle$ and $|\downarrow\rangle$ by themselves represent states that the system can exist in, but we have to be very careful about how we interpret $|\psi\rangle$, because not everyone would agree that the system is "in both the up and down state simultaneously", some would argue that this is overstepping what we can meaningfully interpret physically.
Even the idea of the state "collapsing" upon measurement is in contention as we don't really know what's happening at the point of measurement. This is the area of different interpretations of quantum mechanics which you can find extended discussion about for instance here.
$^1$ We will neglect most of the technicalities here i.e. "it's not really a single vector it's a ray" etc.
