"We are asked to prove:"
\qquad \qquad \qquad \qquad \qquad { 1+ 2 sinx cosx } / { sinx + cosx } \ = \ sinx + cosx.
"Looking at the LHS, we have the following:"
\qquad { 1+ 2 sinx cosx } / { sinx + cosx } \ = \ { ( sin^2x + cos^2x )+ 2 sinx cosx } / { sinx + cosx }
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ = \ { sin^2x + 2 sinx cosx + cos^2x } / { sinx + cosx }
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ = \ ( sinx + cosx )^2 / { sinx + cosx }
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ = \ sinx + cosx.
\qquad :. \qquad \qquad \qquad \qquad { 1+ 2 sinx cosx } / { sinx + cosx } \ = \ sinx + cosx. \qquad \qquad \qquad \qquad \ \ (!)