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

Question

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.

Answer 1

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

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

Answer 2

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

Answer 3

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

Answer 4

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

Answer 5

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

Answer 6

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

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

Answer 7

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

Answer 8

No, they are not equal. Here is why.

We can look at an algebraic approach to arcsec and arccos first:

Arccos(x) = y
Cos(y) = x
1/sec(y) = x
Sec(y) = 1/x
y = arcsec(1/x)

Now, we have established that Arccos(x) = y = arcsec(1/x), Thus, arccos(x) = arcsec(1/x) And we have got a neat algebraic proof.

Conversely, we can do the reverse by substituting cos for sec in the above equations to get arcsec(x) = arccos(1/x)

Take a graphing calculator, you can substitute the values, and this is the image of the graph

Lastly, the longest side of a triangle is 1/cos(x) = y, and since y is not equal to sec(y), the above equation in the question is wrong ( 1/cos(x) is not sec(y), and 1/arccos(x) is not arcsec(x) ).