This expression can be simplified by using some trigonometric identities12. First, we can use the identity sin(2x) = 2sin(x)cos(x) to rewrite the first term. Then, we can use the identity cos(2x) = cos^2(x) - sin^2(x) to rewrite the fourth term. We get:
sin(x) + 2cos(x) - cos(2x) + cos(x) = 2sin(x)cos(x) + 2cos(x) - (cos^2(x) - sin^2(x)) + cos(x) = 2sin(x)cos(x) + 3cos(x) - cos^2(x) + sin^2(x)
Next, we can use the identity sin^2(x) + cos^2(x) = 1 to simplify the last two terms. We get:
= 2sin(x)cos(x) + 3cos(x) - cos^2(x) + sin^2(x) = 2sin(x)cos(x) + 3cos(x) - cos^2(x) + (1 - cos^2(x)) = 2sin(x)cos(x) + 3cos(x) + 1
This is the simplest form of the expression.