Do the components of a force written for a purpose actually exist? On an inclined plane if you put a box, the force of gravity $mg$ is written as sum of two forces $mg\sin\theta$ and $mg\cos\theta$ where $\theta$ is the angle the incline is making with earths surface. Do these forces $mg\sinθ$ and $mg\cosθ$ actually work on the object?
 A: What you are describing is called 'vector decomposition'. Your question is a bit philosophical in nature and there is no straight answer. It is the same as asking "if you are holding 5 apples, are you actually holding 3 apples and 2 apples?" Both statements are true: you are holding 5 apples and you are also holding 3 and 2 apples. If you translate this to your problem you get that the force of gravity ($mg$) is acting on an object but you can also write this as $\hat{\mathbf n}\,mg\cos\theta+\hat{\mathbf t}\,mg\sin\theta$. Here $\mathbf n,\mathbf t$ refer to the normal/tangential directions. They are the same mathematically.
So the answer is technically yes but it's a bit of a confusing way to think about this.
A: 
Do these forces $mg\sinθ$ and $mg\cosθ$ actually work on the object?

Only the component of the force of gravity, $mg\sin\theta$, does work on the object. That work is then
$$W=Fd=mgd\sin\theta$$
Where $d$ is the length of the incline plane traveled by $m$. Since $\sin\theta=h/d$ where $h$ is the height of the plane,
$$W=mgh$$
Which tells us the work done by gravity depends only on the initial and final vertical position of $m$, regardless of the path between the two positions. This is a property of conservative forces, of which gravity is one.
Hope this helps.
A: If you put two force meters on the block, one in direction of the incline, one orthogonal to it the two meters show the two forces you calculated  and stay so if you take away the inclined plain, So the two forces really exist not just in theorie .
A: In the case of no friction, a load cell on the incline surface measures $F_N = mgcos(\theta)$. And the box has an acceleration of $a = gsin(\theta)$, what by the Newton's second law implies a force $F_t = mgsin(\theta)$.
So, the $2$ forces are real in the meaning that they can be measured. Their vectorial sum happens to be $mg$, which could be verified placing the same box over an horizontal load cell surface.
A: I am tempted to say that in the larger picture, given that the force $m\vec g$ comes from interaction with the earth, the components of the vector we are considering simply happen to be in the direction in which something of our interest is happening, the direction of the incline plane in this context, and thus $mgsin\theta$ and $mgcos\theta$ themselves aren't individual forces given that they haven't arisen from different sources.
But as @AccidentalTaylorExpansion mentioned, this is indeed more of a philosophical question that falls in the same sort as asking "If I punch a person and my hand hurts, did they hurt me or did I hurt myself". While the larger source of the action was you, the person being there obviously still was the cause of your hand hurting. At the end of the day, you might or might not treat components as human constructs, but they give mathematical form to the effect some vector quantity has in another direction, so they very much exist.
A: Sort of..?
If you stand on bottom of ramp and you consider the distance between you and block along the incline as a function of time $p(t)$, then you can find that:
$$ \frac{d^2 p}{dt^2} = mg \sin \theta$$
In absence of friction.
