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

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Answer 1 · 2015-06-18T02:47:31

It is not possible to verify #(1+tan(x))/(1+cot(x)) = 2
because it isn't (in general) true.

As an obvious counter example consider the case when $x=pi/4$ .

$color(white)("XXXX")$ $(1+tan(pi/4))/(1+cot(pi/4))$

$color(white)("XXXX")$ $color(white)("XXXX")$ $=(1+1)/(1+1)$

$color(white)("XXXX")$ $color(white)("XXXX")$ $= 1$

$color(white)("XXXX")$ $color(white)("XXXX")$ $!= 2$

In fact with very little manipulation it is easy to show:
$color(white)("XXXX")$ $(1+tan(x))/(1+cot(x)) = tan(x)$

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