I'm supposed to show
$\nabla^\nu\nabla_\nu x^\mu = -g^{\lambda\nu}\Gamma^\mu_{\lambda\nu}$
I think the right approach is to write $\nabla^\nu = g^{\nu\sigma}\nabla_\sigma$ for the first covariant derivative and then just write it all out. However, I can't get things to cancel out which leads me to think I made an elementary mistake somewhere. I also vaguely remember having done a similar question before without a ton of nasty algebra.
I'll appreciate any help! :)
Let $\mu$ be fixed, then
$$\nabla_\nu x^\mu = \delta_\nu^{\ \mu}\Rightarrow \nabla^\nu \nabla_\nu x^\mu = g^{\nu\alpha}\nabla_\alpha \delta_\nu^{\ \mu} =g ^{\nu\alpha} \left( \partial_\alpha \delta_\nu^{\ \mu} - \Gamma_{\alpha \nu}^\beta \delta_\beta^{\ \mu}\right) = -g^{\nu\alpha} \Gamma_{\alpha \nu}^\mu$$