is the unique solution of $\cos t = t$ a transcendental number?

Question

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.