# Is arctan(x) = cot(x) true?

Oct 23, 2015

No.

#### Explanation:

$\arctan \left(x\right)$ is the inverse function of $\tan \left(x\right)$, and it means that, if $y = \arctan \left(x\right)$, then $y$ is a number such that $\tan \left(y\right) = x$.

In general, $f$ is the inverse function of $g$ if $f \left(g \left(x\right)\right) = g \left(f \left(x\right)\right) = x$.

On the other hand, $\cot \left(x\right)$ simply is $\frac{1}{\tan} \left(x\right)$, so it's simply the inverse number of $\tan \left(x\right)$.

So, you have that, as a function, $\arctan \left(x\right)$ is the inverse of $\tan \left(x\right)$, which means that composing the two functions results in the identity function. In formulas,

artan(tan(x))=tan(arctan(x)=x.

Instead, as a number (i.e. you must fix $x$), $\cot \left(x\right)$ is the inverse of $\tan \left(x\right)$, which means that multiplying the two numbers gives one as a result:

$\tan \left(x\right) \cot \left(x\right) = 1$ for every $x$.