I have two sets of complex coordinates $(z_1,z_2)$ and $(t_1,t_2)$. I have the following relation between them $$ \begin{split} z_1&=\frac{t_1}{\sqrt{|t_1|^2+|t_2|^2}}\tan\left(\sqrt{|t_1|^2+|t_2|^2}\right)\\ z_2&=\frac{t_2}{\sqrt{|t_1|^2+|t_2|^2}}\tan\left(\sqrt{|t_1|^2+|t_2|^2}\right) \end{split} $$ I know that $(z_1,z_2)$ are the affine coordinates on the complex projective space $CP^2$ Now I have four one forms $(Y_1,Y_2, Y_3, Y_4)$ they are complicated functions of $(t_1,t_2)$. I want to rewrite the Fubini Study metric on $CP^2$ in terms of the forms$(Y_1,Y_2, Y_3, Y_4)$ . I have been able to workout the following relations $$ \begin{split} \frac{\bar{z}_1 dz_1+\bar{z}_2 dz_2}{1+|z_1|^2+|z_2|^2} &=\bar{z}_1Y_1+\bar{z}_2Y_2\\ \frac{z_1d\bar{z}_1+z_2 d\bar{z}_2}{1+|z_1|^2+|z_2|^2}&=-(z_1Y_3+z_2Y_4) \end{split} $$ I don't know the exact relation between $(z_1,z_2)$ and $(Y_1,Y_2, Y_3, Y_4)$. Is there a way to get the relation using the above condition so that I can rewrite the Fubini-Study metric in terms of these forms?