Where exactly is the following process incorrect to yield an impossible answer

Question

I was playing with my calculator and found some strange phenomena.

$\cos(\tan(\tan(\tan(\pi/4)))) = 0.75686700166$

Verify here

Now when we apply some inverses, then

$\tan(\tan(\tan(\pi/4))) = \arccos(0.75686700166)$

$or, \tan(\tan(\tan(\pi/4))) = 0.712290287$

Verify here

Similarly, applying further arctan

$\tan(\pi/4) = 0.554220248 $

$or, 1 = 0.5544220248$

Now, this conclusion is impossible, so something we assumed must be incorrect! Method of contradiction. But in this case, which step would be considered incorrect or inaccuarate?

Is it just because of the precision of numbers? Or is there something else i am missing here?

Answer 1

Note that $\tan(\tan(\tan(\frac{\pi}4)))= 74.686$

$\cos (74.686)= 0.757$ as you have

$\arccos (0.757) = 0.619$

The multiple tan function takes you out of the interval $[0,2\pi)$