Verifying trigonometric identities, 1 + tan x/ 1 + cot x = tan x?

Answer 1

The answer is Yes!

Numerator = 1 + tanx = 1 + (sinx/cosx) = (cosx + sinx)/cosx
Denominator = 1+ cotx = 1+ (cosx/sinx) = (sinx + cosx)/sinx

Then,

(1+tanx) / (1+cotx) = sinx/cosx = tanx.

Answer 2

trick is to unravel the main complicated section that's clearly the left side (a million + tan x)/ (a million+cot x) substitute cot to a million/tan x (a million+ tan x)/ (a million+ (a million/tan X) (a million+ tan X) / ((tan X +a million)/ tan X) ( a million+ tan x) X tan x/ (tan x + a million) cancel out one yet another and you get tan x = tan x

Answer 3

(1 + tan(x))/(1 + cot(x))
(1 + tan(x))/(1 + (cosx/sinx))
(1 + (sinx/cosx)) / ((sin(x) + cos(x))/sin(x))
((cos(x) + sin(x))/(cos(x)) / ((sin(x) + cos(x))/sin(x))
((cos(x) + sin(x))/cos(x)) * ((sin(x)/(sin(x) + cos(x)))
(sin(x) * (cos(x) + sin(x))) / (cos(x) * (cos(x) + sin(x))
(sin(x))/(cos(x))
tan(x)

so

(1 + tan(x))/(1 + cot(x)) = tan(x)