the expression (1+tanx+tan^2x)(1-cotx+cot^2x) has the positive values for x , given by?

Question

for all real values of x in the domain
HOW??????

Answer 1

If you solve the equation by expanding tan(x) and cot(x):
(1+sin(x)/cos(x) + sin^2(x)/cos^2(x)) (1-cos(x)/sin(x)+ cos^2(x)/sin^2(x))

the end equation is:

(sin(x) + cos(x))^2/2(sin(x)cos(x))^2
as sin and cos are never negative in the first quadrant (one increases from zero and the other decreases from 1), for all positive values they will always have a positive number

Answer 2

(1+tanx+tan^2x)(1-cotx+cot^2x) = 1 - cot x + cot² x + tan x - 1 + cot x + tan² x - tan x + 1 = 1 + cot² x + tan² x

Since tan x and cot x are real numbers, tan² x and cot² x are never negative. Therefore, the expression is always greater than or equal to 1, and therefore always positive.

Answer 3

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