Deduce a related series from an implicit differential equation

The curve $y=f(x)$ passes through $(0,\pi/4)$ and satisfies $y^\prime\tan y=\ln((e^x+1)/2)$. Find the first three nonzero Maclaurin terms of $y$. Deduce the expansion of $(\cot y)\ln((e^x+1)/2)$ through $x^2$.