Repeatedly 'cosine'ing a number: convergence

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ChaoticLlama
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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.
 
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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.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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