TeX source:

\begin{split} \rho_A &= \text{tr}_B(\rho_{AB}) \\ &= \frac{1}{2}\left\{\text{tr}_B(|0_A0_B\rangle\langle0_A0_B|) + \text{tr}_B(|0_A0_B\rangle\langle1_A1_B|) + \text{tr}_B(|1_A1_B\rangle\langle0_A0_B|) + \text{tr}_B(|1_A1_B\rangle\langle1_A1_B|)\right\} \\ &= \frac{1}{2}\left\{\text{tr}_B(|0_A\rangle\langle0_A|\otimes|0_B\rangle\langle0_B|) + \cdots \right\} \\ &= \frac{1}{2}\left\{|0_A\rangle\langle0_A|\langle0_B|0_B\rangle + |0_A\rangle\langle1_A|\langle1_B|0_B\rangle + |1_A\rangle\langle0_A|\langle0_B|1_B\rangle + |1_A\rangle\langle1_A|\langle1_B|1_B\rangle\right\} \\ &= \frac{1}{2}\left\{|0_A\rangle\langle0_A| + |1_A\rangle\langle1_A|\right\} = \frac{1}{2}\mathbf{I}. \end{split}