Is $\pi$ equal to $180^\circ$?

$$ \begin{array}{ccc} \sin{(\theta+180^{\circ})}=-\sin{\theta} & \cos{(\theta+180^{\circ})}=-\cos{\theta} & \tan{(\theta+180^{\circ})}=\tan{\theta} \\ \sin{(\theta+\pi)}=-\sin{\theta} & \cos{(\theta+\pi)}=-\cos{\theta} & \tan{(\theta+\pi)}=\tan{\theta} \end{array} $$

If I compare them, I will get $\pi=180^{\circ}$. Why? Isn't $\pi=3.142\ldots $? Can anyone prove this?

Not $\pi$ but $\pi$ radians equal $180°$

It would be reasonable to define that: $$x^\circ = \frac{2\pi}{360}\cdot x$$ in which case yes, $180^\circ$ literally equals $\pi$. Leox's answer is probably a little more correct though.

Addendum. After a bit of thought, I've changed my mind slightly; I no longer think that Leox's answer is correct anymore. To summarize my current beliefs about the issue: $\pi$ literally equals $180^\circ$, both are unitless (as others have argued), and neither degrees nor radians are really units at all (again, as others have argued.) In particular, I think that "radians" and "degrees" are basically systems of conventions, not units like meters or seconds.

Lets discuss this a little. In my opinion, what's really going on is that there is a function

$$\mathrm{AngleInRadians} : \mathbb{R}^2 \times \mathbb{R}^2 \rightarrow [0,\pi]$$

given by

$$\mathrm{AngleInRadians}(v,w) = \mathrm{arccos}\left(\frac{v \cdot w}{\|v\| \cdot \|w\|}\right)$$

and another function,

$$\mathrm{AngleInDegrees} : \mathbb{R}^2 \times \mathbb{R}^2 \rightarrow [0,180]$$

given by

$$\mathrm{AngleInDegrees}(v,w) = \frac{180}{\pi}\mathrm{arccos}\left(\frac{v \cdot w}{\|v\| \cdot \|w\|}\right)$$

Observe that both functions return unitless numbers. So really, degrees and radians aren't units at all; they're not like meters or seconds. They're more like consistent systems of conventions than anything.

If we want to formalize the relationship between these conventions, then $x^\circ$ should be defined as stated in my original answer, as the result of evaluating a function $(-)^\circ : \mathbb{R} \rightarrow \mathbb{R}$ at a (unitless) number $x.$ Explicitly:

$$(-)^\circ : \mathbb{R} \rightarrow \mathbb{R}$$

$$x^\circ = \frac{\pi}{180} \cdot x.$$

It follows that:

$$\mathrm{AngleInRadians}(v,w) = (\mathrm{AngleInDegrees}(v,w))^\circ.$$

Under this convention, statements like $\pi = 180^\circ$ and $\cos(\theta+180^\circ) = -\cos \theta$ are literally true, where $\cos$ is viewed as a mathematical function $\mathbb{R} \rightarrow \mathbb{R}$. So the inputs to $\cos$ are mere numbers; they have no units, and neither do its outputs.