To find the value of cos8x 2cos6x cos4x, we need to use the given equation sinx sin2x = 1 and apply trigonometric identities to simplify the expression. Let's break it down step by step:

Step 1: Simplify sinx sin2x = 1
Using the double angle identity for sine, we can rewrite sin2x as 2sinxcosx:
sinx 2sinxcosx = 1

Step 2: Rearrange the equation
To simplify further, let's rearrange the equation:
2sinxcosx + sinx = 1

Step 3: Factor out sinx
In order to factor out sinx, we can rewrite the equation as follows:
sinx(2cosx + 1) = 1

Step 4: Solve for sinx
Dividing both sides of the equation by (2cosx + 1), we get:
sinx = 1/(2cosx + 1)

Step 5: Simplify cos8x 2cos6x cos4x
Now, let's substitute sinx with the value we found in step 4 into the expression cos8x 2cos6x cos4x:
cos8x 2cos6x cos4x = cos8x 2cos6x (1 - sin^2(2x))

Step 6: Apply trigonometric identities
Using the Pythagorean identity sin^2(2x) + cos^2(2x) = 1, we can rewrite the expression as:
cos8x 2cos6x (1 - sin^2(2x)) = cos8x 2cos6x cos^2(2x)

Step 7: Apply double angle identities
Using the double angle identities for cosine, we can rewrite the expression as:
cos8x 2cos6x cos^2(2x) = cos8x 2cos6x (2cos^2(2x) - 1)

Step 8: Simplify further
Expanding the expression, we have:
cos8x 2cos6x (2cos^2(2x) - 1) = 2cos8xcos^2(2x) - cos8x - 4cos6xcos^2(2x) + 2cos6x

Step 9: Apply trigonometric identities again
Using the double angle identity for cosine, we can rewrite the expression as:
2cos8xcos^2(2x) - cos8x - 4cos6xcos^2(2x) + 2cos6x = 2cos8x(1 + cos(4x))/2 - cos8x - 4cos6x(1 + cos(4x))/2 + 2cos6x

Step 10: Simplify and combine like terms
Simplifying further, we have:
cos8x(1 + cos(4x)) - cos8x - 2cos6x(1 + cos(4x)) + 2cos6x = cos8x + cos8xcos(4x) - cos8x -