Question

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

Answer 1

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.