# 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.

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Go to wolframalpha.com/widgets/… and input log10(tan(x) + sec(x)) as your fuction. –  Matthew Hoggan May 15 '14 at 21:53
en.wikipedia.org/wiki/Arcsecant –  MattAllegro May 15 '14 at 21:53
–  nadia-liza May 15 '14 at 21:53
Remember that arcsec delivers an angle as its value. Ordinarily, it makes no sense to take the reciprocal of an angle. –  Lubin May 15 '14 at 22:58

No. If you graph $\sec^{-1}(x) \cdot \cos^{-1}(x)$, you get:

You can clearly see that it isn't $1$.

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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 $$\boxed{\sec^{-1} x = \cos^{-1}\tfrac1x}$$

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

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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

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

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

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

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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.

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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 $$\bbox[4pt, border:1px solid blue;]{\color{red}{\sec^{-1}(x)=\cos^{-1}\left(\frac{1}{x}\right)}}$$

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