Trigonometry:

Simplify the expression.

(cot²α - 4) / (cot²α - cotα - 6)

(cot²α - 4) / (cot²α - cotα - 6); Let x = cotα

Therefore (cot²α - 4) / (cot²α - cotα - 6) = (x² - 4) / (x² - x - 6)

(x² - 4) = x² - 2² = (x-2) (x+2). Difference of two squares.

(x² - x - 6); This is a quadratic expression,

Multiply the first and last terms = - 6 x²

we think of two expression that multiply to give - 6x² and add up to give - x (Middle term)

Those expression are 2x and - 3x

(x² - x - 6) = (x² + 2x-3x - 6) = x (x+2) - 3 (x+2) = (x+2) (x-3)

Recall (x² - 4) / (x² - x - 6) = ((x-2) (x+2)) / ((x+2) (x-3)) = (x-2) / (x-3). Cancelling out.

Recall x = cotα, therefore:

(x-2) / (x-3) = (cotα-2) / (cotα-3)

(cot²α - 4) / (cot²α - cotα - 6) = (cotα-2) / (cotα-3)