MHB Which is Greater? $\cos(\cos 1))$ vs $\cos(\cos(\cos 1)))$

[anemone]

Determine which of the following is greater?

$\cos(\cos 1))$ or $\cos(\cos(\cos 1)))$

 

[kaliprasad]

Determine which of the following is greater?

$\cos(\cos 1))$ or $\cos(\cos(\cos 1)))$

Spoiler

we have $\cos 1 < 1$
so $\cos(\cos 1) > (\cos 1)$
so $\cos(\cos(\cos 1)) < \cos (\cos 1))$
Hence $2^{nd}$ term is larger

above is using the fact that $\cos{t} < 1 < \frac{\pi}{2}$ and for $a > b$ $\cos (a) < \cos (b) $

 

[Opalg]

Just to illustrate kaliprasad's solution (from a dynamical systems perspective):
[sp]The brown curve is $y=\cos x$, the blue line is $y=x$.
View attachment 5480
[/sp]

 

[anemone]

Thanks kaliprasad for your solution! I solved it the same way you did!

A special thanks to Opalg for attaching the graph for illustration purpose! I appreciate it!