Which fixed point iteration will converge? and why? Which of the following fixed point iterations will converge, and why? Also if possible please give the rate of convergence. 
(a) $x_{n+1}=\cos(x_n)$
(b) $x_{n+1}=\sin(x_n)$
(c) $x_{n+1}=\tan(x_n)$
Thank you!
 A: As explained in this answer (as well as in many sources on functional iteration), a fixed point $x_0$ is attractive if $\left|f'(x_0)\right|<1$ and repulsive if $\left|f'(x_0)\right|>1$. This explains your observation on the cosine and provides the rate of convergence, namely $\left|f'(x_0)\right|$. The tangent clearly has a lot of fixed points where the slope is larger than one that are all repulsive.  
The tricky questions involve the behavior of the sine and the tangent at the $x_0=0$, which is a neutral fixed point for both.  By neutral, I mean that $\left|f'(x_0)\right|=1$, which implies that the dynamics near this point are very slow.  Thus, numerical experimentation might be unreliable.
If I iterate the sine function $100,000$ times starting from $x_0=1.0$, I find that the last few terms are
\begin{align}
x_{99997}&=0.00547705\\
x_{99998}&=0.00547702\\
x_{99999}&=0.00547700\\
x_{100000}&=0.00547697
\end{align}
Note that the terms are decreasing.  This must happen since $0<\sin(x)<x$, whenever $x>0$. As a result, this sequence is decreasing and bounded below, so it mush have a limit.
Experimentation suggested these results, but does not provide proof.  If we iterate the tangent function $100,000$ starting from $1.0$, we get something like the following:
\begin{align}
x_{99997}&=-0.00233827\\
x_{99998}&=-0.00233828\\\
x_{99999}&=-0.00233828\\
x_{100000}&=-0.00233829
\end{align}
Staying close to zero, so maybe convergent? Well, no - they're getting larger in absolute value.  After $200,000$ iterates, we're only at about $-0.0029332$ but moving away.  Can you see why?
I should also probably mention that the exact value of $100,000$ iterates are highly suspect, particularly in the presence of repulsive orbits.  Nonetheless, the basic types of behavior can be gleaned from this type of experimentation.
