To simplify the expression (sinx + sin3x) / (cosx + cos3x), we can use the sum-to-product identities for sine and cosine functions.

First, rewrite the expression as:

sinx/cosx + sin3x/cos3x

Now, apply the sum-to-product identity sin(A) / cos(B) = tan(A-B):

tan(x) + tan(3x)

Now, we can simplify tan(3x) using the triple angle formula for tangent:

tan(3x) = (3tan(x) - tan^3(x)) / (1 - 3tan^2(x))

Therefore, the final simplified expression is:

tan(x) + (3tan(x) - tan^3(x)) / (1 - 3tan^2(x))