When proving them the "modern" way (from first principles) it seems impossible to get around proving the identities $$\lim_{x\to 0}\frac{\sin x}{x} = 1$$ and the related $\cos$ limit. This itself requires some geometric reasoning and uses the definition of radians that we have today.
But surely people knew that $(\sin \theta)'=\cos \theta$ and so forth long before limits came about. Is this just obvious? How would they have justified it?
Did you try looking in any books on the history of calculus? The following is taken from "The Historical Development of the Calculus" by C. H. Edwards (p. 205 ff).
The inverse sine function (for radian angles in the first quadrant) can be related to an area under an arc of the unit circle, which is $y = \sqrt{1-x^2}$ in the first quadrant. Newton knew from his discovery of the binomial expansion for rational exponents how to write $\sqrt{1-x^2}$ as a power series, which he could then integrate termwise. In this way he found the power series for $\arcsin x$. Then he inverted that series until he recognized the pattern to "establish" the power series of $\sin x$, from which he could find the power series for $\cos x$ as the series for $\sqrt{1-\sin^2x}$ with constant term $1$. From the power series expansions of $\sin x$ and $\cos x$ it is clear that $\sin'x = \cos x$.
The power series for $\sin x$ and $\cos x$ were known in India long before they were found by Newton. See https://en.m.wikipedia.org/wiki/Madhava_series.
While the power series formulas reveal the derivatives of the sine and cosine to us, it is not the case that Newton or Leibniz in the 1600s took this step. The trigonometric functions were not systematically thought of as functions to be graphed and have a derivative. Euler, in the 1700s, was the one who first incorporated the trigonometric functions into calculus. See "The calculus of the trigonometric functions" by Victor Katz, which is available online at http://ac.els-cdn.com/0315086087900644/1-s2.0-0315086087900644-main.pdf?_tid=a15b6d9e-9c37-11e5-b17b-00000aacb35e&acdnat=1449420025_6f4ed309a5aed67668cf37b14c276280.
Knowledge of the specific fact that $(\sin x)' = \cos x$ actually predates the general knowledge of calculus and derivatives. It was known in the following form: that for very small $\Delta x$, when you increase $x$ to $x + \Delta x$, the increase in value of the sine, from $\sin x$ to $\sin (x + \Delta x)$, is proportional to $\Delta x$ times $\cos x$. In other words, that $$\frac{\sin (x + \Delta x) - \sin x}{\Delta x} \approx \cos x$$ The approximation being exact in the limit as $\Delta x \to 0$ is of course the modern definition of the derivative.
This happened historically in Indian mathematics, where Muñjala (around 932), Āryabhaṭa II (around 950), Prashastidhara (around 958) all give the above rule for calculating $\sin(x + \Delta x)$, and an explicit geometric reasoning / justification is given by Bhāskara II (around 1150) in his Siddhanta Shiromani. I have not found a perfectly good reference to these, but you can start with the following article:
It was first pointed out by Bapu Deva Shastri in Bhaskara's knowledge of the Differential Calculus, Journal of the Asiatic Society of Bengal, Volume 27, 1858, pp. 213–6.
Formally, $(\sin x)'=\cos x$ is true only if you measure x in radians. So you need to use radians for this fact anyway. If you will to ignore this, you can say that Euclid knew that the limit of $\frac{\sin x}{x}$ is 1, at least when $x=2\pi/n$ and $n\rightarrow \infty$. Of course Euclid did not use radians, sin and lim, but he proved that the difference between the area of a circle and the area of an inscribed right polygon will become arbitrarily small as number of sides becomes large (by method of exhaustion). The area of right n-gon inscribed in a circle of radius 1 is $\frac 12 n\sin (2\pi/n)$. So using our notation, we can say that Euclid proved that $$\lim_{ n\rightarrow \infty} \frac 12 n\sin (2\pi/n) = \pi.$$ This is equivalent to $$\lim_{n\rightarrow \infty}\frac{ \sin (2\pi/n)}{2\pi/n} = 1.$$ Btw, that was used by Archimed to estimate $\pi$: he approximated $\pi$ by area of 96-gon. That is, he calculated $\sin(2\pi/96)$ and used that $2\pi/96 >\sin(2\pi/96)$. And to prove the upper bound he used $x< \tan x$.
I guess the initial discovery and subsequent justification may have been done just by trying some calculations, like below. Nowadays we can use Excel for this kind of investigation.
