Why Riemannian metrics defining the same angles are conformal

Question

Suppose $g_1$ and $g_2$ are two metrics defining the same angles, which means $g_1(X,Y)/(g_1(X,X)g_1(Y,Y))^{0.5}=g_2(X,Y)/(g_2(X,X)g_2(Y,Y))^{0.5}$ for all pairs of vector $X,Y$.I want to prove that $g_1=cg_2$ for a constant $c$. I have tried to use the polarization identity，but it doesn't work.