Let $K$ be a local field. Let $n$ be a natural number which is not divisible by the residue characteristic and consider the maximal unramified extension $K_{nr}.$ I would like to show that $H^1(Gal(K_{nr}/K),\mu_n) \cong U_K K^{*n}/K^{*n}$ where $\mu_n$ are the $n$th roots of unity in $K_{nr}.$
I thought I had a proof of this, but I'm a bit unsure since I seem to have two different proofs giving two different results.
Consider the SES $$0 \rightarrow \mu_n \rightarrow K_{nr} \rightarrow K_{nr}^{*n} \rightarrow 0.$$ Looking at the LES of this, I am led to suspect that $H^1(Gal(K_{nr}/K),\mu_n) \cong K^*/K^{*n}$ which is not correct.
On the other hand, using the inflation-restriction exact sequence $$1 \rightarrow H^1(Gal(K_{nr}/K),\mu_n) \rightarrow H^1(Gal(K^{sep}/K),\mu_n) \rightarrow H^1(Gal(K^{sep}/K_{ur}),\mu_n)$$ I seem to get that $H^1(Gal(K_{nr}/K),\mu_n) \cong \mu(K)/\mu(K)^n$ and this seems to be isomorphic to $U_K K^{*n} /K^{*n}.$ This since $U_K K^{*n}/K^{*n} \cong U_K /U_K^{n} \cong \mu(K) / \mu(K)^n$ (here we're using the structure of the unit group of local fields).
What exactly is my mistake in the first approach?