# Does $\operatorname{arcsec}(x) = 1 /\arccos(x)$?

Does $\operatorname{arcsec}(x) = 1 /\arccos(x)$? I have looked in a few books and Google'd it but I am not finding my answer.

No. If you graph $\sec^{-1}(x) \cdot \cos^{-1}(x)$, you get:  You can clearly see that it isn't $1$.

If $\sec^{-1} x = \theta$, then $x = \sec\theta$. This means $\frac1x = \cos\theta$, so $\cos^{-1}\frac1x = \theta$. So your equation is wrong; the correct statement is $$\boxed{\sec^{-1} x = \cos^{-1}\tfrac1x}$$

• $$(\sec x)^{-1} = (\cos x)^1 \\ \sec^{-1} x = \cos^{-1} x^{-1}$$ it’s kinda satisfying . . . feels like the negative ones cancel each other out – gen-z ready to perish Apr 26 '19 at 21:18

Actually it's: $$\operatorname{arcsec}(x)=\arccos(1/x).$$

No. It is not. If you look at the definitions

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

and then we solve for the x

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

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

and replace $x$ and $y$ to find the inverse

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

• Careful, $\cos^{-1}(\cos(x)) \neq x$ in some cases. – recursive recursion May 17 '14 at 18:49

Isn't. Draw $$\text{arcsec} x\arccos x$$

• Please avoid link only answers and add a plot if you are telling the OP to test his theory by graphing. – Cole Johnson May 16 '14 at 17:03

No, it is false. Probably you meant $\operatorname{arcsec}(x)=\arccos(1/x)$, which is true.

It is straight forward, let $$\sec^{-1}(x)=\alpha$$ $$\implies \sec\alpha=x$$ $$\implies \frac{1}{\cos \alpha}=x$$$$\implies \cos \alpha=\frac{1}{x}$$ $$\implies \alpha=\cos^{-1}\left(\frac{1}{x}\right)$$ Substituting the value $\alpha=sec^{-1}(x)$, we get $$\bbox[4pt, border:1px solid blue;]{\color{red}{\sec^{-1}(x)=\cos^{-1}\left(\frac{1}{x}\right)}}$$