find solutions with trigonometric identities

The first step is to use trigonometric identities to simplify this expression. One of the most basic trig identities you will ever need to memorize is:

sin^2x + cos^2x = 1

Now, we can manipulate this to find a way to simplify cot^2x by dividing both sides by sin^2x to get this:

sin^2x/sin^2x + cos^2x/sin^2x = 1/sin^2x

1 + cot^2x = csc^2x

cot^2x = csc^2x - 1

Now, let's put this back into the original problem:

csc^2x - 1 + cscx - 1 = 0

csc^2x + cscx - 2 = 0

This looks crazy, but it's really just a quadratic equation. Let's simplify it by letting u = cscx. You just get:

u^2 + u - 2 = 0

Which can be factored into:

(u + 2)(u - 1) = 0

Therefore, this is true for the values of u:

u = -2, and u = 1

Since u = cscx, this means that:

cscx = -2, and cscx = 1

Or:

sinx = -1/2, and sinx = 1

Assuming your answer needs to be between 0 and 360 degrees, taking the arcsin of these values gives:

x = 210, 330 (for the first relation), and x = 90 (for the second relation) degrees (multiply by pi/180 to get radians).