I'm reading Tong's notes on GR http://www.damtp.cam.ac.uk/user/tong/gr.html and i cannot understrand how he derived the equation that relates the varation of the metric with its inverse in page 141 under equation 4.2.
A different approach could be that $$g_{\mu\nu}g^{\mu\nu}=d \to \delta g_{\mu\nu}g^{\mu\nu} = -\delta g^{\mu\nu}g_{\mu\nu}$$ where $d$ is the spacetime dimension, but i cannot understand his calculation.
Your approach won't work. How are you to isolate the $\delta g^{\mu \nu}$? You only have one scalar equation!
In the notes, we have schematically $AB = D$ where $D$ is a constant matrix and $B$ is the inverse to $A$. Now take the variation $(\delta A) B + A (\delta B) = 0$ (if you prefer think of this as equations for each component). Now we can isolate $\delta B$ by multiplying by the inverse of $A$ (which is $B$), so $B (\delta A) B + \delta B = 0$.
