# Inequality involving $\ln(\sin(x))$

I'm interested by the following problem :

Let $$x,y>0$$ such that $$x<\dfrac{\pi}{4}$$, $$y<\dfrac{\pi}{4}$$ then we have : $$\ln(\sin\Bigg(\frac{\frac{x}{\tan(x)}+\frac{y}{\tan(y)}}{\frac{1}{\tan(x)}+\frac{1}{\tan(y)}}\Bigg))+\ln(\sin \bigg(\frac{x+y}{2} \bigg))\leq \ln(\sin(x))+\ln(\sin(y))$$

Since $$\ln(\sin(x))$$ is concave with the condition I would apply the Jensen's inequality but it's too weak so I'm a bit lost . If you have a hint it would be nice . Thanks a lot for your time .