given is $\displaystyle y(x)=ln(tan(\frac{x}{2}))-atan(x)ln(1+sin(x))-x$

so I would like to derivate this

thus we can split it in 3 part

First $\displaystyle ln(tan(\frac{x}{2}))$ wil be $\displaystyle \frac{1}{2}\frac{sec^2(\frac{x}{2})}{tan(\frac{x}{ 2})}$

the second and third one $\displaystyle atan(x)ln(1+sin(x))-x$ will be $\displaystyle \frac{ln(1+sin(x))}{sin^2(x)}-\frac{atan(x)cos(x)}{1+sin(x)}-1$

finelay we get $\displaystyle \frac{1}{2}\frac{sec^2(\frac{x}{2})}{tan(\frac{x}{ 2})} - \frac{ln(1+sin(x))}{sin^2(x)}-\frac{atan(x)cos(x)}{1+sin(x)}-1$

Who can I simplify this? Greets Thanks a lot.