Sin^2x+6cos^2x+7sinxcosx=0

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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.