$\mathrm{Sin}2\mathrm{\theta }+\frac{1}{3}{\mathrm{Sin}}^{3}2\mathrm{\theta }+\frac{1}{5}{\mathrm{Sin}}^{5}2\mathrm{\theta }+\dots \dots \infty =$

Given eqn

$$\begin{gathered} =\frac{1}{2} \log \left(\frac{1+\sin \mathrm{\theta }}{1-\sin \mathrm{\theta }}\right) \\ =\frac{1}{2} \log {\left(\frac{\cos \mathrm{\theta }+\sin \mathrm{\theta }}{\cos \mathrm{\theta }-\sin \mathrm{\theta }}\right)}^2 \\ =\log \left(\frac{1+\tan \mathrm{\theta }}{1-\tan \mathrm{\theta }}\right) \\ =\log \tan \left(\frac{\mathrm{\pi }}{4}+\mathrm{\theta }\right) \end{gathered}$$