How to solve algebraically the equation $x = \frac{1}{2}\cos\left(\frac 2 3 \sin\left(\frac 3 4 x\right)\right) + 1$ 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 ?

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 ($=0.5772156649...$). 
