# Are cotangent and negative tangent graphs the same?

##### 1 Answer

No. See explanation.

#### Explanation:

Let's look at the definitions in terms of sines and cosines (because tangents and cotangents are just ratios of sines and cosines).

Cotangent of x is:

and negative tangent of x is:

Cotangent of x equals 0 when the numerator

This happens when

On the other hand, tangent (or negative tangent) of x becomes 0 when the numerator

This happens when

**Therefore, the two equations are not the same.**

You can also see this, by assuming the two equations are equal, and finding a contradiction when making this assumption.

Let's try this.

If the two equations were the same, then their difference should be equal to 0.

We write that statement as such:

(Note: the sign becomes a + because we're doing

Then we rewrite in terms of sines and cosines:

which simplifies to:

(Note: I used the identity

**You see now that this is never true for all x, and therefore, that this equality is false.**

This means that our assumption ("the two equations are equal") is wrong, **which means our two equations are not equal.**

You can see it in the following graphs too.

This is

graph{cot x [-10, 10, -5, 5]}

and this is

graph{-tan x [-10, 10, -5, 5]}

Q.E.D.