If we want to calculate $\sin(x)$ and $\cos(x)$ for very large $x$ , lets say $10^5$ , the usual way is to reduce the number $x$ modulo $2\pi$.
If the number is a large power of a small number, for example $2^{200}$, we could also use $$\cos(2x)=\cos^2x-\sin^2x$$ and $$\sin(2x)=2\sin(x)\cos(x)$$ multiple times.
Do we have any chance, if the power is too high for both methods ? For example, can we calculate $\sin(x)$ and $\cos(x)$ for $x=10^{10^{10^{10}}}$ ?
Note, that the power tower must be calculated from above. So, we have
$$x=10^{(10^{(10^{10))}}}$$
In tetration-notation, we can write $x=10\uparrow\uparrow 4$
By computing $\cos(x)$ and $\sin(x)$ (and thus $\exp(ix)$), you are in effect computing the fractional part of $x/(2\pi)$. Any numerical procedure that attempts to do this (approximately) must deal with the fact this is extremely sensitive to small relative errors in $x$, when $x$ is large. Thus if you want to approximate $\cos\left(10^{10^{10}}\right)$ to any accuracy, you'd need to do computations with at least $10^{10}$ significant digits. This might be near the boundary of feasibility with today's computers. But for $\cos\left(10^{10^{10^{10}}}\right)$, you'd need $10^{10^{10}}$ significant digits; the known universe wouldn't be big enough to store that many digits.