How do you prove (1+tan x) / (1+cot x) = 2?

1 Answer
Feb 19, 2016

It is not an identity and thus, it can't be proven.

Explanation:

You can't prove it since it's not an identity.

Let's transform the left side using

tan x = sin x / cos x, " " cot x = cos x / sin x

We have

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

= (cos x/ cos x + sin x / cos x) / (sin x / 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)

= (cancel((cos x + sin x)) * sin x) / (cos x * (cancel(sin x + cos x)))

= sin x / cos x

= tan x

However, tan x = 2 is certainly not true for all x.

Thus, your equation is not an identity and can't be proven.