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?
 A: YES. They are the same.
The are the same in the sense that $12  = 1\ \textrm{dozen}$.

Remember, that an angle is a ratio. Is is the ratio of the length of the arc to the radius. For a circle, that ratio is $2\pi$. For a half circle, that ratio is $\pi$.
If I said that the angle (the ratio) was $0.75$, or $3/4$, or $\textrm{three quarters}$, those are exactly the same.
"Radians" doesn't mean anything, other than the number is being applied to a angle, rather than some other ratio or number. It can always be left off and be perfectly valid.
"Degrees" is a quantity of units, similar to $\textrm{quarter}$ or $\textrm{dozen}$. The only difference is that one degree is not a rational number of units.
Just as you could say 5 dozen mph, you could say 3437.7... degrees mph.
Of course, no one measures in dozens mph, just as no one measures in degrees mph.
But they could.
A: Not $\pi$ but $\pi$  radians equal $180°$  
A: 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.
A: For most branches of mathematics, the most useful definition of angle is the ratio between arc length and radius. As such, the angle is a quotient of lengths and hence dimensionless. To indicate that a certain number is meant as an angle in this sense, you can affix the unit “radian”, but that's just for clarification and therefore often not written down.
On the other hand, for most engineering purposes, the radian is a terrible unit of angle, and the degree is a much more useful one. So you might want to convert between these units. To do so you'd say “$180° = \pi\,\text{rad}$”. But that's the engineering point of view, where you treat “rad” as a unit. If you take the mathematical view, you read that unit simply as one and indeed end up with $180°=\pi$.
You might even go one step further, interpret the $180°$ as a product between the number $180$ and the constant (or dimensionless unit) $°=\pi/180$. In this fashion, things like $x°$ meaning $x\times\frac{\pi}{180}$ have a well-defined meaning as well. You can even plot the graph of some function (e.g. $\sin(x)$), label the ticks on one axis using plain numbers $30,60,90,\dots$ and label that whole axis to represent $x/°$. Which can correctly interpreted as “$x$ divided by that constant” but is usually better read as “$x$ measured in degrees”.
A: This started as a comment, but got rather long.
Degrees and radians are two ways of relating the size of angles. Angles are dimensionless - geometrically similar figures have the same angles regardless of size. However the units with which an angle is measured do make a difference - essentially it comes down to what fraction of the whole circle is represented by the number $1$.
A division into $360$ pieces goes with measuring time in minutes and seconds. The scale factor $2\pi$ relates to the intrinsic geometry of the circle (the circumference is $2\pi$ times the radius), and also turns out to avoid some inconvenient constants in differential and integral calculus of the trigonometric functions.
Though dimensionless, the scale of measurement of angles also comes into particular focus when analysing circular motion, or motion in polar co-ordinates. Again it turns out that the radian scaling leads to formulae without inconvenient constants.
A: Your $\tan$, $\cos$ and $\sin$ examples do not prove anything, even if you precede them with $\forall \theta$.
No one really uses $^\circ$ in any professional mathematics. If you were to define it in some reasonable terms then you need to follow what goblin wrote. And this definition explicitly uses $\pi$. Therefore, yes $180^{\circ} = \pi$.
A: $2\pi$ is equal to one turn, but since that is an impractical irrational number, what we do is to take a nice number with a lot of divisors like $360^\circ$ and use it to measure one turn. In this way, instead of using crazy fractions of $\pi$, we can call many common angles with nice integer numbers, but do not forget the $^\circ$!
Basically: $\pi\equiv180^\circ \neq180$.
Note that in the very same way you can define: $2\pi\equiv400^g$, this unit actually exists and is called gradian.
A: Strictly speaking, there are two functions that are commonly denoted by $\sin(\cdot)$.
The mathematical sine function, $\sin: \mathbb R \to \mathbb R,$
has the real numbers as its domain.
That is, the function does not take angles to numbers; it takes numbers to numbers.
The domain of the other sine function is angle measurements;
an angle measurement consists of a number and the units of measurement of the angle.
Just as a single interval of times or a single linear distances can be written in multiple
different ways using various numbers with various units,
a single angle can be written in multiple different ways with different units.
Hence it is correct to write
$$ 180^\circ = \pi\mbox{ rad},$$
where the symbol $^\circ$ indicates units of degrees and rad is the symbol for radians,
or (if you are in a context where it is appropriate
to use the "other" function named $\sin(\cdot)$)
to write
$$ \sin(30^\circ) = \sin\left(\frac\pi6  \mbox{ rad}\right).$$
On the other hand, in a more "pure" mathematical context using the function
$\sin: \mathbb R \to \mathbb R,$
strictly speaking we should write
$$\sin\left(\frac\pi6\right) = \frac 12 \neq \sin(30) \approx -0.988.$$
In practice, the tendency to interpret the notation $\sin(30)$ as
$\sin(30^\circ)$ is so strong that if you type sin(30) as input to
Wolfram Alpha (for example) it will return $0.5$ as the answer.
On the other hand if you put =sin(30) in a cell in some widely-used
spreadsheet programs you may be in for a surprise.
One just has to be aware of this potential source of confusion 
(identical names for two different functions) and make sure one uses the
correct function in the given context.
A: Obviously, $\pi$ is treated as a real number. While in trigonometric functions, it is treated as an angle in radian which is exactly equal to $180^o$. 
Note: $\sin(\pi-\theta)=sin\theta$ Here, $\pi (\text{radian})=180^o$
Also note in simple calculations, $\pi-180\approx -176.8584073...$ 
Here, $\pi$ is treated as a real number ($\pi\approx 3.141592654$)      
A: This image below shows how to interpret Degrees and Radians. A full-circle= $360^\circ$.
Quoting directly from wikipedia:

Radian describes the plane angle subtended by a circular arc as the length of the arc divided by the radius of the arc. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. More generally, the magnitude in radians of such a subtended angle is equal to the ratio of the arc length to the radius of the circle; that is, θ = s /r, where θ is the subtended angle in radians, s is arc length, and r is radius. Conversely, the length of the enclosed arc is equal to the radius multiplied by the magnitude of the angle in radians; that is, s = rθ.
As the ratio of two lengths, the radian is a "pure number" that needs no unit symbol, and in mathematical writing the symbol "rad" is almost always omitted. When quantifying an angle in the absence of any symbol, radians are assumed, and when degrees are meant the symbol ° is used.

Now, if you were to equate Radians and Degrees,then:
$$2\pi  \text {   radians}=360^\circ$$
$$ 1\text {   radian}=\frac{180^\circ}{\pi}\approx 57.3^\circ$$

Source : Lucas V. Barbosa on Wikimedia Commons and Tumblr
A: There are two different ways of measuring angles, both using a circle centered on the vertex of the angle:


*

*The fraction of the circle it cuts off.  This is traditionally measured in degrees, where $1^\circ$ is $1/360$th of a circle.  So e.g. a right angle is $90^\circ$.

*The distance to travel along the unit circle to get from one edge to the other.  So e.g. a right angle is $1/4 \times \text{the circumference of the unit circle}$.


The key point is that we almost always unify these, by defining$
\newcommand{\calc}{\begin{align} \quad &}
\newcommand{\op}[1]{\\ #1 \quad & \quad \unicode{x201c}}
\newcommand{\hints}[1]{\mbox{#1} \\ \quad & \quad \phantom{\unicode{x201c}} }
\newcommand{\hint}[1]{\mbox{#1} \unicode{x201d} \\ \quad & }
\newcommand{\endcalc}{\end{align}}
\newcommand{\ref}[1]{\text{(#1)}}
$ $$
\tag{0} 1^\circ \;=\; 1/360 \times \text{the circumference of the unit circle}
$$
And since
$$
\tag{1} \pi \;=\; 1/2 \times \text{the circumference of the unit circle}
$$
we trivially have
$$\calc
180^\circ
\op=\hint{by $\ref{0}$}
180/360 \times \text{the circumference of the unit circle}
\op=\hint{by $\ref{1}$}
\pi
\endcalc$$
(Note. Since $\;\text{the circumference of the unit circle}\;$ plays such an important role in trigonometry, the symbol $\;\tau\;$ is used as an abbreviation.  More traditionally, textbooks still use $\;2\pi\;$.)

On radians (which the OP did not ask about, but everyone apparently wants to talk about): "2 radians" means "a distance of 2 traveled around the unit circle, to measure an angle".  So the word 'radians' is there to describe the intent of the number, viz. to measure an angle.
As an analogy: I can say, "From my home to the office is 10 miles as the crow flies."  The phrase "as the crow flies" does not change the distance of 10 miles, nor does it change the unit: it describes the intent, to make sure the reader is not confused.
A: 
If I compare them, I will get $\pi = 180^{\circ}$. Why? 

Beyond the obvious fact that: 
It takes $3.14\dots$ or $\pi$ lengths of a radius of a circle1 to rotate a point on it on $180^{\circ}$, i.e. the angle subtended at the centre of a circle is expressed as a length of an arc with units of measurement the radius of that same circle.  
I will address a side that is not covered a lot, namely, why exactly $180^{\circ}$ an not any other number (conveniently skipping the irrational to rational "convertion" part)?
The use of the scale1 from $0$ to $360$ for measuring degrees of angles is probably related to the number system2 of the people that first started describing angles, viz. partly for historical reasons, partly for convenience reasons.
My guess is that in the used scale $[0,360]$, $180$ should be thought of merely as an approximation of $\pi$.

1. Which is a interval numeric scale.
2. Obviously not decimal but more likely the Babylonian Sexagesimal,i.e. with base 60 as it is easier to calculate fractions like 1/2, 1/4 etc.

