To simplify this expression, we can use the Pythagorean trigonometric identity:
sin^2(x) + cos^2(x) = 1
First, let's replace sin^2(x) with (1-cos^2(x)):
(1-cos^2(x)) + 6cos^2(x) + 7sin(x)cos(x) = 0
Now, let's combine like terms:
1 - cos^2(x) + 6cos^2(x) + 7sin(x)cos(x) = 0
Simplify further:
1 + 5cos^2(x) + 7sin(x)cos(x) = 0
Now, let's apply the double angle formula for sine:
7sin(x)cos(x) = 7(1/2)sin(2x) = (7/2)sin(2x)
Substitute this back into the equation:
1 + 5cos^2(x) + (7/2)sin(2x) = 0
This is the simplified expression for the given trigonometric equation.
To simplify this expression, we can use the Pythagorean trigonometric identity:
sin^2(x) + cos^2(x) = 1
First, let's replace sin^2(x) with (1-cos^2(x)):
(1-cos^2(x)) + 6cos^2(x) + 7sin(x)cos(x) = 0
Now, let's combine like terms:
1 - cos^2(x) + 6cos^2(x) + 7sin(x)cos(x) = 0
Simplify further:
1 + 5cos^2(x) + 7sin(x)cos(x) = 0
Now, let's apply the double angle formula for sine:
7sin(x)cos(x) = 7(1/2)sin(2x) = (7/2)sin(2x)
Substitute this back into the equation:
1 + 5cos^2(x) + (7/2)sin(2x) = 0
This is the simplified expression for the given trigonometric equation.