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 $(Y_1,Y_2, Y_3, Y_4)$. Also note that $Y_3=-\bar{Y}_1,~Y_4=-\bar{Y}_2$. 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?
Special Case: To make my question clear let us consider the case $z2=0$. Then we see the relationship reduces to the following
$$
\begin{split}
\frac{ dz_1}{1+|z_1|^2} &=Y_1\\
\frac{d\bar{z}_1}{1+|z_1|^2}&=-Y_3
\end{split}
$$
Taking the product of the above equation gives
$$
\begin{split}
ds^2=\frac{ dz_1d\bar{z}_1}{(1+|z_1|^2)^2}=-Y_1Y_3=Y_1\bar{Y}_1
\end{split}
$$
This is the Fubini Study metric on $CP^1$ with inhomogeneous/affine coordinates $(z_1,z_2)$. I am expecting something like this for the case of $CP^2$ also.