homework and exercises - How to calculate Riemann and Ricci tensors for a sphere?

Let's have the metric for a sphere: $$ dl^{2} = R^{2}\left(d\psi ^{2} + sin^{2}(\psi )(d \theta ^{2} + sin^{2}(\theta ) d \varphi^{2})\right). $$ I tried to calculate Riemann or Ricci tensor's components, but I got problems with it.

The Ricci curvature must be $$ R_{ij}=\frac{2}{R^{2}}g_{ij}. $$ But when I use definition of Ricci tensor, I can't turn the expression into the expression for the metric tensor

Maybe, there are siome hints, which can help?