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

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The discussion revolves around comparing the values of $\cos(\cos 1)$ and $\cos(\cos(\cos 1))$. Participants utilize a dynamical systems perspective, highlighting the behavior of the function $y=\cos x$ against the line $y=x$. Graphical illustrations are provided to clarify the solution, with specific thanks given to contributors for their insights. The consensus indicates a collaborative effort to understand the comparative values through graphical representation. Ultimately, the discussion emphasizes the importance of visual aids in solving mathematical comparisons.
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Determine which of the following is greater?

$\cos(\cos 1))$ or $\cos(\cos(\cos 1)))$
 
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anemone said:
Determine which of the following is greater?

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

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) $
 
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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]
 

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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!
 
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