If you are talking about getting the $90 \times 10 = 900$ correlation coefficients between what you call the dependent and independent variables, this is only a matter of using a statistical software.
If you are using R, it would be like this
# Suppose X is a nx90 matrix with independent variables
# and that Y is a nx10 matrix with the dependent variables.
cor(X,Y) gives a $90 \times 10$ matrix with the coefficients you are looking for. Below I give an example with fake data but real matrices so that you can check what the output looks like.
# Say n = 1000.
X <- matrix(rnorm(90*1000), ncol=90)
Y <- matrix(rnorm(10*1000), ncol=10)
But honestly, 900 coefficients of correlation will not lead you very far without further analysis. Perhaps you are actually searching for the linear combination of
X that best correlates with a linear combination of
Y, or trying to fit a 10-dimensional multivariate response on 90 predictors?
Intuitively, I'd say you are actually aiming for a canonical correlation analysis, of which you can find a primer on Jeromy Anglim's blog. In R it looks like this.
cca <- cancor(X,Y)
# Highest correlation between pairs of linear combinations
# Corresponding linear combination in X (90 coefficients)
# Corresponding linear combination in Y (10 coefficients)