I am aware that according to Lindemann–Weierstrass theorem:
1) $\sin(a),\;\cos(a),\;\tan(a)$, and their multiplicative inverses $\csc(a),\;\sec(a),$ and $\cot(a)$, for any nonzero algebraic number $a$, the result will be transcendental.
2) $\ln(a)$ if $a$ is algebraic and not equal to $0$ or $1$ will also be transcendental.
However, if you put any value (except $1$) for $x$ in $\cos(\ln(x))$, according to Wolfram Alpha, the result will be transcendental.
I would like to know why that is so. Does anyone have a proof that every value of that function will be transcendental? (if that is the case)
Best Answer
$\exp(i \ln(x)) = x^i$ (for a suitable branch). Since $i$ is algebraic and not rational, the Gelfond-Schneider theorem says that $\exp(i \ln(x))$ is transcendental whenever $x$ is an algebraic number other than $0$ or $1$. And from that you can get that $\cos(\ln(x))$, $\sin(\ln(x))$, $\tan(\ln(x))$ are also transcendental.