Inverse Secant

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The function $\sec^{-1}x$ , where $\sec x$ is the Secant and the superscript denotes the Inverse Function, not the multiplicative inverse. The inverse secant satisfies

$\begin{displaymath} \sec^{-1}x=\csc^{-1}\left({x\over\sqrt{x^2-1}}\right) \end{displaymath}$

(1)

for Positive or Negative

, and

$\begin{displaymath} \sec^{-1}x=\pi-\sec^{-1}(-x) \end{displaymath}$

(2)

for $x\geq 0$ . The inverse secant is given in terms of other inverse trigonometric functions by

$\displaystyle \sec^{-1}x$	$\textstyle =$	$\displaystyle \cos^{-1}\left({1\over x}\right)$	(3)
	$\textstyle =$	$\displaystyle \cot^{-1}\left({1\over\sqrt{x^2-1}}\right)$	(4)
	$\textstyle =$	$\displaystyle {\textstyle{1\over 2}}\pi-\csc^{-1}x=-{\textstyle{1\over 2}}\pi-\csc^{-1}(-x)$	(5)
	$\textstyle =$	$\displaystyle \sin^{-1}\left({\sqrt{x^2-1}\over x}\right)$	(6)
	$\textstyle =$	$\displaystyle \tan^{-1}(\sqrt{x^2-1}\,)$	(7)

for $x\geq 0$ .

See also Inverse Cosecant, Secant

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 141-143, 1987.