TITLE: is the unique solution of $\cos t = t$ a transcendental number?
QUESTION [8 upvotes]: let $\alpha$ be the unique fixed point of $\cos:\mathbb{R} \rightarrow [-1,1]$
for any $t \in \mathbb{R} \setminus\{0\}$ if $t$ is algebraic then $\cos t$ is transcendental. thus if $\alpha$ were algebraic it must also be transcendental, a contradiction, since $\cos\alpha = \alpha$. hence $\alpha$ is transcendental.
QUESTION is this argument valid?
NOTE re the stated assumption, in answer to this recent question  Prahlad Vaidyanathan pointed me to  this Wikipedia entry re the Lindemann-Weierstrass theorem.

REPLY [2 votes]: Sifting throuht the material of https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem one concludes that 
$\alpha \ne 0$ and algebraic $\implies e^{\alpha}$ transcendental
Assume that $\cos t = t$ is algebraic. Then $it$ is also algebraic and $\ne 0$, and so is $s=e^{it}$, as the solution of the equation $\frac{s+1/s}{2} = t$, contradiction with the above for $\alpha = i t$.
Yes, the argument of the OP is valid.