Is $\cos\log a$ a transcendental for all nonzero algebraic $a$?

Question

Is this known? (excluding a=1 as was corrected in comments)

Answer 1

Assuming that $a\neq 1$ is a positive algebraic number, $$\cos\log a = \frac{e^{i\log a}+e^{-i\log a}}{2} = \frac{a^i+a^{-i}}{2} $$ is trascendental for the same reason that grants that $e^{\pi}$ is trascendental: the Gelfond-Schneider theorem.