The graph of y=cos x * ln cos^2x has seven horizontal tangent lines on the interval [0,2pi]. Find the x-coordinate of all points at which these tangent lines occur.

To find the derivative of the function, we can use the product rule and chain rule.

Let's break down the function into two parts:
1. y = cos(x)
2. g(x) = ln(cos^2(x))

Now, let's find the derivative of each part individually:

1. For y = cos(x), the derivative is dy/dx = -sin(x)

2. For g(x) = ln(cos^2(x)), we can use the chain rule. Let u = cos^2(x), then g(x) = ln(u), and the derivative is dg/du * du/dx:

dg/du = 1/u
du/dx = -2sin(x)cos(x) (using the chain rule on u = cos^2(x))

So, the derivative of g(x) with respect to x is dg/dx = dg/du * du/dx = (1/u) * (-2sin(x)cos(x)) = -2sin(x)cos(x) * (1/cos^2(x)) = -2tan(x)

Now, using the product rule, the derivative of the function y = cos(x) * g(x) is:

dy/dx = (cos(x) * (-2tan(x))) + (-sin(x) * g(x))

Simplifying this expression, we get:

dy/dx = -2sin(x)tan(x) - sin(x)ln(cos^2(x))

Therefore, the derivative of the function y = cos(x) * ln(cos^2(x)) is dy/dx = -2sin(x)tan(x) - sin(x)ln(cos^2(x)).

Now that we have the derivative, we can proceed to the next part of the question.