Question

For what values of `x` does `tan(-2x) = cot(x)` ?

Answer 1

`x = ((2k+1)pi)/2" "` for any integer `k`.

Explanation:

Given:

`tan(-2x) = cot(x)`

That is:

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

Multiplying both sides by `cos(-2x)sin(x)`, this becomes:

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

Subtracting `sin(-2x)sin(x)` from both sides, we get:

`cos(x)cos(-2x)-sin(x)sin(-2x) = 0`

Compare the left hand side with the sum formula for `cos`:

`cos(alpha)cos(beta)-sin(alpha)sin(beta) = cos(alpha+beta)`

So with `alpha=x` and `beta=-2x` we get:

`cos(-x) = 0`

Note that `cos(-x) = cos(x)`

Hence:

`x = ((2k+1)pi)/2`

where `k` is any integer.

Here are the two functions `tan(-2x)` and `cot(x)` plotted together:

graph{(y+tan(2x))(y-cot(x)) = 0 [-10, 10, -5, 5]}