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How to solve this trigonometric equation $x = \frac 1 2 \cos\left(\frac 2 3 \sin\left(\frac 3 4 x\right)\right) + 1$ ?

The iterative solution seems to be 1.417.

Can anybody suggest an algebraic solution ? Does it really exist ?

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    In fact, Schanuel implies much more, the algebraic independence of $\pi$ and $e$, which Timothy Chow was able to extend to show that no solution to $x = \cos x$ can be expressed in closed form (as precisely defined in Chow's paper). See Theorem 1 on p. 443 of Chow's paper. A URL for a freely available version is .2011-07-14

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You can easily get extremely accurate approximations using (for example) Wims function calculator, by searching the root $x^*$ of $ f(x) = x - \bigg[\frac{1}{2}\cos \bigg(\frac{2}{3}\sin \bigg(\frac{3}{4}x\bigg)\bigg) + 1\bigg] $ (say between $-5$ and $5$). Computing its value up to a mild number of digits gives $ x^* = 1.417520004.... $ While Inverse symbolic calculator does not recognize this, it does lead to the approximation $ x^* \approx \frac{{10 \sin (1)}}{{e^{\exp (\gamma )} }} = 1.417520089... , $ where $\gamma$ is Euler's constant ($=0.5772156649...$).