Graph of Composite Sine and Cosine Function I was playing around with a graphing calculator and I noticed that if you repeatedly apply sines and cosines in this manner:
$$\sin(\cos(...{\sin (\cos{(\sin(x)}))))}$$
its graph flattens out. Why is this so?

There is an important idea in functional analysis called a contraction map. A contraction map is a function $f$ such that there is a positive number $a<1$ such that for any two distinct points $x$ and $y$, $$\left|\frac{f(x)-f(y)}{x-y}\right|<a.$$ If $f$ is differentiable, then you can think of this as the slope of $f$ is always less than $a$, which is in turn less than $1$. The importance of contraction maps comes from the fact that we re guaranteed the existence of a unique fixed point $x$ such that $f(x)=x$. Moreover, starting from any point $y$, the sequence $\{f(y),f(f(y)),f(f(f(y))),...\}$ converges to $x$. So if $\cos(\sin(x))$ is a contraction map, then it makes sense that repeatedly applying this function would pull all points close to the unique fixed point, making the graph almost constant. To see that $\cos(\sin(x))$ is a contraction map, we take a derivative:
$$\frac{d}{dx}\cos(\sin(x))=-\sin(sin(x))\cos(x).$$ 
Though it is hardly a proof, a quick graph of the derivative should be enough to convince you that the absolute value of the derivative is less than $.5$. The mean value theorem then gives us that your function is a contraction map.
