To solve the equation cos^2x - 3sin xcosx = -1, we can first use the Pythagorean identity sin^2x + cos^2x = 1 to rewrite cos^2x as 1 - sin^2x:

(1 - sin^2x) - 3sin xcosx = -1
1 - sin^2x - 3sin xcosx = -1

Next, we can factor out a sin x from the last two terms:

1 - sin^2x - 3sin xcosx = -1
1 - sin^2x - 3sin xcosx = -1
1 - sin^2x - sin x(3cosx) = -1

Then, we can use the double angle formula cos(2x) = 1 - 2sin^2(x) to rewrite -3cosx as -3(1 - 2sin^2(x)):

1 - sin^2x - sin x(3cosx) = -1
1 - sin^2x - sin x(3(1 - 2sin^2(x))) = -1
1 - sin^2x - 3sin x + 6sin^3(x) = -1

Now we can rearrange the equation to get everything on one side:

1 - sin^2x - 3sin x + 6sin^3(x) + 1 = 0

Finally, we can notice that this is a cubic equation in terms of sin(x), and we can try to solve for sin(x) using numerical methods or a cubic equation solver.