Question

Trigonometry: Simplify the expression. (cot²α - 4) / (cot²α - cotα - 6)

Answer 1

(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 = -6x²
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)