Question

How do you simplify `(2)/(1/tan x-tan x)` ?

`(2)/(1/tan x-tan x)`

Answer 1

I'm going to guess what you mean by simplify and say that this expression simplifies down to `tan(2x)`.

Explanation:

(I am assuming that this problem is about rewriting a complicated trigonometric expression in different form; often this is done as an exercise of proving identities).

So we start with the expression
`2/(1/tan x - tan x)`

Using the identity that `1/tan x = cot x`

`2/(cot x - tan x)`

Since `cot x = cos x/sin x` and `tan x = sin x/cos x`:

`2/((cos x/sin x)-(sin x/cos x)`

We can give this a common denominator.

`2/((cos^2x-sin^2x)/(cos x sin x))`

Since dividing by a fraction is the same as multiplying by its reciprocal

`(2sinxcosx)/(cos^2x-sin^2x)`

Now if we remember the double angle formulas `sin(2x)=2sinxcosx` and `cos(2x)=cos^2x-sin^2x`

`sin(2x)/cos(2x) = tan(2x)`