Repeatedly 'cosine'ing a number: convergence

[ChaoticLlama]

A curiosity I've had recently...

When repeatedly taking the cosine of a number in radians, it appears to converge to a value.

i.e. cos(cos(cos(...cos(x)...))) = 0.73908513321516064165531208767387...

Any thoughts/explanations/exact solution etc?

Thanks.

 

[Treborman]

Yes, this value does converge to the solution to the equation cos(x) = x, because the input, x, must be the same as the output, cos(x), if it converges.