= Solution
Let
$$
U=X_1+X_2,\qquad V=X_1-X_2.
$$
They are jointly normal because they are linear combinations of a Gaussian <vector>. When $\sigma_1=\sigma_2=\sigma$,
$$
\operatorname{cov}(U,V)
=\operatorname{var}(X_1)-\operatorname{var}(X_2)=0.
$$
Hence <uncorrelated jointly normal variables are independent>. Their means and variances are
$$
\mathbb EU=\mu_1+\mu_2,
\qquad
\operatorname{var}U=2\sigma^2(1+\rho),
$$
and
$$
\mathbb EV=\mu_1-\mu_2,
\qquad
\operatorname{var}V=2\sigma^2(1-\rho).
$$
Therefore
$$
\boxed{U\sim N(\mu_1+\mu_2,2\sigma^2(1+\rho))},
$$
$$
\boxed{V\sim N(\mu_1-\mu_2,2\sigma^2(1-\rho))},
$$
independently.
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