$$y=\ln {\sin }^{2}x$$

$$g=si{n}^{2}x \Rightarrow y=\ln g$$

$$u=sin x ; u\text{'}=cos x\Rightarrow g=u^2 \Rightarrow g\text{'}=2·u·u\text{'}=2·sin x·cos x$$

$$y\text{'}=\frac{1}{g}·g\text{'}$$

$$y\text{'}=\frac{1}{{\sin }^{2}x}·2·sinx·\cos x$$

$$y\text{'}=\frac{2\cos x}{\sin x}$$