To solve this trigonometric equation Cos(2x) = sin(x-90) - 1, we first need to use the double angle identity for cosine:
Cos(2x) = 2 * Cos^2(x) - 1

Substitute this into the equation:
2 * Cos^2(x) - 1 = sin(x-90) - 1

Now, we need to use the trigonometric identities to simplify the equation. Remember that sin(x-90) = sin(x) cos(90) - cos(x) sin(90) = sin(x) 0 - cos(x) 1 = -cos(x)

Therefore, the equation simplifies to:
2 * Cos^2(x) - 1 = -cos(x) - 1

Rearranging the terms, we get:
2 * Cos^2(x) + cos(x) = 0

Now, we have a quadratic equation in terms of cosine. Let's set it to zero and solve for cosine:
2 Cos^2(x) + cos(x) = 0
Cos(x) (2 Cos(x) + 1) = 0

So, either Cos(x) = 0 or 2 * Cos(x) + 1 = 0

Cos(x) = 0
This happens when x = 90 degrees, or x = 270 degrees.

2 Cos(x) + 1 = 0
2 Cos(x) = -1
Cos(x) = -1/2

The solutions to this are x = 120 degrees and x = 240 degrees.

Therefore, the solutions to the trigonometric equation Cos(2*x) = sin(x-90) - 1 are x = 90 degrees, x = 120 degrees, x = 240 degrees, and x = 270 degrees.