# Inverse reciprocal trig functions

#### otownsend

Hi,

I hope someone can help. Can someone please explain to me why sec^1(x) does NOT equal 1/cos^1(x) and why sec^1(x) does equal cos^1(1/x)?

The notation can get a bit confusing.

Thanks,
Olivia

#### Ahri

Do you know how to find the inverse of a function $f(x)$?

Write $$f(x) = \frac{1}{\cos(x)}=y$$

Interchange the $x$ and $y$ to get

$$\frac{1}{\cos(y)}=x$$

Isolate $y$,

$$\cos(y)= \frac{1}{x}$$
$$y= \cos^{-1} \left(\frac{1}{x}\right)$$

Which is $f^{-1}(x)$.

#### otownsend

Okay yeah I understand that. Thank you