Why do we require radians in calculus? I think this is just something I've grown used to but can't remember any proof.
When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does it work then and only then?
 A: Assume that the formula $\sin'_r(x)=\cos_r(x)$ is true for some angular unit, let "$r$". For another angular unit, let "$d$", there is a conversion factor, let $\lambda_{d\rightarrow r}$, and we can write:
$$\sin_d'(x)=\sin_r'(\lambda_{d\rightarrow r} x)=\lambda_{d\rightarrow r}\cos_r(\lambda_{d\rightarrow r} x)=\lambda_{d\rightarrow r}\cos_d(x).$$
So the derivation formula can only be simple ($\lambda=1$) for the specific angular unit $r$, which we use to call radians.
But how do we know how much is a radian ?
Using $\sin_r'(x)=\cos_r(x)$ (and in turn $\cos_r'(x)=-\sin_r(x)$) allows to derive various Taylor-McLaurin series expansions, among which that of the arc tangent, and eventually leads to the Gregory-Leibnitz formula. This defines the constant $\pi$ and shows that an eighth of a turn (angle of the isosceles right triangle) is $\frac{\pi}{4}$ radians, equivalent to 45 degrees (by definition of the degrees).
In the end, $\lambda_{d\rightarrow r}=\frac{\pi}{180}$ and $\sin_d'(x)=\frac{\pi}{180}\cos_d(x)$.
A: Simple answer: radian is not really a unit, it's an absence of one. Degree, on the other hand, is. Working with dimensionful quantities in calculus is the last thing you would want to do (unless you're into kinky things) :)
A: 
Radians make it possible to relate a linear measure and an angle
  measure. A unit circle is a circle whose radius is one unit. The one
  unit radius is the same as one unit along the circumference. Wrap a
  number line counter-clockwise around a unit circle starting with zero
  at (1, 0). The length of the arc subtended by the central angle
  becomes the radian measure of the angle.

From Why Radians? | Teaching Calculus
We are therefore comparing like with like the length of a radius and and the length of an arc subtended by an angle $L = R \cdot \theta$ where $L$ is the arc length, $R$ is the radius and $\theta$ is the angle measured in radians.
We could of course do calculus in degrees but we would have to introduce awkward scaling factors.
The degree has no direct link to a circle but was chosen arbitrarily as a unit to measure angles: Presumably its $360^o$ because 360 divides nicely by a lot of numbers. 
A: To make commenters' points explicit, the "degrees-mode trig functions" functions $\cos^\circ$ and $\sin^\circ$ satisfy the awkward identities
$$
(\cos^\circ)' = -\frac{\pi}{180} \sin^\circ,\qquad
(\sin^\circ)' = \frac{\pi}{180} \cos^\circ,
$$
with all that implies about every formula involving the derivative or antiderivative of a trig function (reduction formulas for the integral of a power of a trig function, power series representations, etc., etc.).

Added: Regarding Yves Daoust's comment, I read the question, "Why does it work [if angles are taken in radians] and only then?", as asking, "Why do the derivative formulas for $\sin$ and $\cos$ take their familiar form when (and only when) $\sin$ and $\cos$ are $2\pi$-periodic (rather than $360$-periodic)?" If this interpretation is correct, and if one accepts that one full turn of a circle is both $360$ units of one type (degrees) and $2\pi$ of another (radians), then the above formulas are equivalent to $\sin' = \cos$ and $\cos' = -\sin$, and (I believe) do justify "why" we use the $2\pi$-periodic functions $\cos$ and $\sin$ in calculus rather than $\cos^\circ$ and $\sin^\circ$.
Of course, it's possible naslundx was asking "why" in a deeper sense, i.e., for precise definitions of "cosine and sine in radians mode" and a proof that $\cos' = -\sin$ and $\sin' = \cos$ for these functions. 
To address this possibility: In my view, it's most convenient to define cosine and sine analytically (i.e., not to define them geometrically), as solutions of the second-order initial-value problems
\begin{align*}
\cos'' + \cos &= 0 & \cos 0 &= 1 & \cos' 0 = 0, \\
\sin'' + \sin &= 0 & \sin 0 &= 0 & \sin' 0 = 1.
\end{align*}
(To say the least, not everyone shares this view!) From these ODEs, it's easy to establish the characterization:
$$
y'' + y = 0,\quad y(0) = a,\ y'(0) = b\quad\text{iff}\quad
y = a\cos + b\sin.
$$
One quickly gets $\cos' = -\sin$ and $\sin' = \cos$, the angle-sum formulas, power series representations, and periodicity (obtaining an analytic definition of $\pi$). After this, it's trivial to see $\mathbf{x}(\theta) = (\cos \theta, \sin \theta)$ is a unit-speed parametrization of the unit circle (its velocity $\mathbf{x}'(\theta) = (\sin\theta, -\cos\theta)$ is obviously a unit vector). Consequently, $\theta$ may be viewed as defining a numerical measurement of "angle" coinciding with "arc length along the unit circle", and $2\pi$ units of this measure equals one full turn.
A: In Calculus, sine and cosine are defined via the exponential function, meaning that 
$ \cos x = \mathrm{Re}\{e^{ix}\}$  and $ \sin x = \mathrm{Im}\{e^{ix}\}$ and as you know, $e^0 = e^{i2\pi} = 1$ which means that $360^{\circ}$ which is the full circle corresponds to $2\pi$. For further reference see here.
A: It is similar to why SI units are used for scientific calculations.
Many people will notice some quantity which they want to measure, for example length or angle. They will come up with repeatable ways of measuring these, designed to be easy to perform. For example the cubit and the foot are based on body parts. Degrees were chosen because they make division easy for many numbers (this is closely related to our measurement of time). The Babylonians used such number systems extensively.
After a long time, some pattern may be spotted. For example, we can calculate the energy used to move something by multiplying the force applied over the distance travelled. This lets us say things like "Energy in calories is proportional to a force in pounds multiplied by a distance in feet", but it doesn't give us an equation. To turn this into an equation requires a constant of proportionality, which will usually be difficult to remember because the units were chosen arbitrarily.
Later on, some bright spark will create a new unit, defined by these patterns. For example, in SI units the energy in Joules is equal to the force in Newtons multiplied by the distance in metres. The constant is 1 by design.
The same is true of radians. Degrees get horrible numbers everywhere because 360 divisions is arbitrary. We can make the equations nicer by changing the number of divisions to a pattern, like Tau (the circumference of a circle / its radius). A system with Tau divisions, instead of 360, makes all of the equations nicer, just like SI units do. That system is Radians.
Note that Tau = 2 * Pi, since Pi is circumference / diameter and diameter = 2 * radius.
A: Another perspective, in my opinion the most "proper" one:
degree is just a named numerical constant that equals ${}^\circ=\frac{\pi}{180}$. So when you read $180^\circ$ you are actually multiplying by that constant! In a similar fashion, conversion into degrees just means that you divide & multiply by a degree and carry out the division, leaving the multiplication unevaluated. In that sense, the degree is just like SI prefixes (kilo, mega, etc.). It just makes the notation more convenient, but it ultimately has no deeper meaning.
The trig functions themselves are, as all other functions, defined for pure number arguments. For instance, through power series $\sin x=\sum\frac{(-1)^n x^{2n+1}}{(2n+1)!}$, and there are no degrees implied in these formulas, it doesn't even imply an angle. It is just a function, it takes any numerical argument.
"Angles" may also appear in formulas outside trig functions (also common in physics). In that case, you cannot survive simply by saying that you used special degree-versions of the trigonometric functions. For instance, the function
$${\rm sinc}(x)=\frac{\sin x}{x} $$
take both $x$ as pure numbers.
A: It really comes down to the following limit:
$$
\lim_{x\to 0} \frac{\sin(x)}{x} = 1
$$
Or in other words, "$\sin x \approx x$ for small $x$".  As a consequence, we have
$$
\frac{d}{dx}\sin x = \cos x, \qquad
\frac{d}{dx}\cos x = -\sin x
$$
For any other choice of angular unit, these derivatives require some sort of coefficient (such as $\pi/180$).  In this sense, radians are the "natural" unit for an angle, as far as calculus is concerned.
A: Here is a rationale admittedly guided by where we want to end up, which is not to say that it isn't natural. Warm-up: From the origin I have a unit vector $(x,y)$ and a second unit vector $(x',y')$ perpendicular to the first and counterclockwise from it. Use geometry to show that $y'=x$ and $x'=-y$ hint: From perpendicularity we have slopes $y'/x'=-x/y.$ 
OK. We want to analyze uniform circular motion so  we  fix a time unit and distance unit. A particle moves with variable position vector $(x,y)$ and associated velocity vector $(x',y'). $ Fix the center at the origin and the two vectors to have length $1$. (aka "move counterclockwise about the unit circle at a linear velocity of $1$ unit/sec".) Then we have$x,y,x',y'=\cos{\theta},\sin{\theta},\cos'{\theta},\sin'{\theta}$ where $\theta$ is the angle moved. The derivative identities? We did that in the warm-up. (And you thought the primes were just decorations , tricky!) Q What is the angular velocity? A: 1 rad/sec. 
An optional digression: if we are trying to develop calculus from first principles then there seems an issue of "circularity" in that the linear velocity involved accumulating arc length and that already requires a fair amount of calculus . A solution (see Apostol, I think) is to instead accumulate sector area. 
A: \begin{align}
\lim_{x\to0} \frac {\sin x} x & = (\text{some constant that is not 0}) \\[12pt]
\frac d {dx} \sin x & = \Big( (\text{some constant that is not 0}) \cdot \cos x \Big)
\end{align}
The "constants" are equal to $1$ if, but only if, radians are used.
A: You can't remember any proof because 'we require radians in calculus' is not a logically well formulated statement, and only those have proofs.
The problem with using degrees in your calculations is that when you integrate or differentiate a function you probably left many logical gaps, which you fill with geometric intuition, which is sometimes misleading.
Differentiating and integrating is a mathematical operation that has nothing to do with the warping you do to the input and/or output of your function so that it takes and returns a range of 'nice' values. This 'warping' MUST be made after all calculations are done, with the only purpose of formatting the data. The problem is that you have $g_1(f(g_2(x)))$ instead of $f(x)$.
Remember: You MUST always first have the analytic definition, then derive the 'geometric interpretation' which is usually just a bunch of analityc properties. Therefore angles must be first defined analytically. The easiest way do it is by the anlytic definitions of sine and cosine, or of the complex exponential if you like. In no part of this process nobody mentions the word 'degree' or 'radian', even though the functions are perfectly differentiable and integrable.
I hope my answer helped.
