TITLE: Taking repeated sin then cos
QUESTION [6 upvotes]: When you keep taking alternating sin and cos of any number as follows:

$$\sin(\cos(\sin(\cos(\sin(\cos...(N))))...)$$

it seems to converge at about 0.69. Is there any way to find the exact value it converges at?

REPLY [2 votes]: If taking sin and cos repeatedly has a limit, call it x. Then $\sin(\cos(\sin(\cos...)))=x$, and so, taking sin and cos of both sides,
$\sin(\cos(\sin(\cos(\sin(\cos...)))))=\sin(\cos(x))$. Since the wo left sides. The quations are equal, so are the right sides.
This can be made rigorous if neccessary.