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Since we defined 9 different inverse trigonometric functions for our 6 trigonometric functions, there are 108 compositions which can be created. When the graphs are inspected and compared, it will be seen that there are 74 different compositions. On this page, we display all of the different compositions and their graphs. We begin with the 54 compositions where the inverse trigonometric function is the inside function.
The first 9 compositions result in three graphs, all of which algebraically simplify to the identity function, whose graph is a straight line. Each such composition is an illustration of the general formula $f(f^{-1}(x))=x$.
| Compositions: | $\begin{array}{ll} \sin(\arcsin x) \\ \cos(\arccos x) \end{array}$ | $\begin{array}{ll} \tan(\arctan x) \\ \cot(\operatorname{ arccot}_1 x) \\ \cot(\operatorname{ arccot}_2 x) \end{array}$ | $\begin{array}{ll} \sec(\operatorname{ arcsec}_1 x) \\ \sec(\operatorname{ arcsec}_2 x) \\ \csc(\operatorname{ arccsc}_1 x) \\ \csc(\operatorname{ arccsc}_2 x) \end{array}$ |
| Simplified: | $x$ | $x$ | $x$ |
| Domain: | $[-1,1]$ | $(-\infty,\infty)$ | $(-\infty,-1] \cup [1,\infty)$ |
| Range: | $[-1,1]$ | $(-\infty,\infty)$ | $(-\infty,-1] \cup [1,\infty)$ |
| Graph: |  |
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The next 18 compositions result in 8 graphs that are portions of conic sections, or quadratic plane curves. Each such composition algebraically simplifies. Of these eight graphs, one is a semicircle, and the other seven are portions of hyperbolas.
| Compositions: | $\begin{array}{ll} \sin(\arccos x) \\ \cos(\arcsin x) \end{array}$ | $\begin{array}{ll} \tan(\operatorname{ arccot}_1 x) \\ \tan(\operatorname{ arccot}_2 x) \\ \cot(\arctan x) \end{array}$ | $\begin{array}{ll} \sec(\arccos x) \\ \csc(\arcsin x) \end{array}$ | $\begin{array}{ll} \cos(\operatorname{ arcsec}_1 x) \\ \cos(\operatorname{ arcsec}_2 x) \\ \sin(\operatorname{ arccsc}_1 x) \\ \sin(\operatorname{ arccsc}_2 x) \end{array}$ |
| Simplified: | $\sqrt{1-x^2}$ | $\dfrac{1}{x}$ | $\dfrac{1}{x}$ | $\dfrac{1}{x}$ |
| Domain: | $[-1,1]$ | $(-\infty,0) \cup (0,\infty)$ | $[-1,0) \cup (0,1]$ | $(-\infty,-1] \cup [1,\infty)$ |
| Range: | $[0,1]$ | $(-\infty,0) \cup (0,\infty)$ | $(-\infty,-1] \cup [1,\infty)$ | $[-1,0) \cup (0,1]$ |
| Graph: |  |
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| Compositions: | $\begin{array}{ll} \tan(\operatorname{ arcsec}_2 x) \\ \cot(\operatorname{ arccsc}_2 x) \end{array}$ | $\begin{array}{ll} \tan(\operatorname{ arcsec}_1 x) \\ \cot(\operatorname{ arccsc}_1 x) \end{array}$ | $\begin{array}{ll} \sec(\arctan x) \\ \csc(\operatorname{ arccot}_2 x) \end{array}$ | $\csc(\operatorname{ arccot}_1 x)$ |
| Simplified: | $\sqrt{x^2-1}$ | $\pm\sqrt{x^2-1}$ | $\sqrt{x^2+1}$ | $\pm\sqrt{x^2+1}$ |
| Domain: | $(-\infty,-1] \cup [1,\infty)$ | $(-\infty,-1] \cup [1,\infty)$ | $(-\infty,\infty)$ | $(-\infty,\infty)$ |
| Range: | $[0,\infty)$ | $(-\infty,\infty)$ | $[1,\infty)$ | $(-\infty,-1) \cup [1,\infty)$ |
| Graph: |  |
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The next 27 compositions result in 15 graphs that are portions of quartic plane curves. Each such composition algebraically simplifies. Each graph is a subset of a graph of the family of equations $x^2 y^2=a_1 x^2+a_2 y^2 +a_3$, where one of the variables $a_i$ has value zero, another of the variables has value one, and the third has value either positive or negative one.
| Compositions: | $\begin{array}{ll} \sec(\operatorname{ arccsc}_1 x) \\ \csc(\operatorname{ arcsec}_1 x) \end{array}$ | $\begin{array}{ll} \sec(\operatorname{ arccsc}_2 x) \\ \csc(\operatorname{ arcsec}_2 x) \end{array}$ | $\sec(\operatorname{ arccot}_1 x)$ |
| Simplified: | $\dfrac{|x|}{\sqrt{x^2-1}}$ | $\dfrac{x}{\sqrt{x^2-1}}$ | $\dfrac{\sqrt{x^2+1}}{|x|}$ |
| Domain: | $(-\infty,-1) \cup (1,\infty)$ | $(-\infty,-1) \cup (1,\infty)$ | $(-\infty,0) \cup (0,\infty)$ |
| Range: | $(1,\infty)$ | $(-\infty,-1) \cup (1,\infty)$ | $(1,\infty)$ |
| Graph: |  |
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| Compositions: | $\begin{array}{ll} \sec(\operatorname{ arccot}_2 x) \\ \csc(\arctan x) \end{array}$ | $\begin{array}{ll} \tan(\operatorname{ arccsc}_1 x) \\ \cot(\operatorname{ arcsec}_1 x) \end{array}$ | $\begin{array}{ll} \tan(\operatorname{ arccsc}_2 x) \\ \cot(\operatorname{ arcsec}_2 x) \end{array}$ |
| Simplified: | $\dfrac{\sqrt{x^2+1}}{x}$ | $\dfrac{\pm 1}{\sqrt{x^2-1}}$ | $\dfrac{1}{\sqrt{x^2-1}}$ |
| Domain: | $(-\infty,0) \cup (0,\infty)$ | $(-\infty,-1) \cup (1,\infty)$ | $(-\infty,-1) \cup (1,\infty)$ |
| Range: | $(-\infty,-1) \cup (1,\infty)$ | $(-\infty,0) \cup (0,\infty)$ | $(0,\infty)$ |
| Graph: |  |
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| Compositions: | $\cos(\operatorname{ arccot}_1 x)$ | $\begin{array}{ll} \sin(\arctan x) \\ \cos(\operatorname{ arccot}_2 x) \end{array}$ | $\begin{array}{ll} \tan(\arcsin x) \\ \cot(\arccos x) \end{array}$ |
| Simplified: | $\dfrac{|x|}{\sqrt{x^2+1}}$ | $\dfrac{x}{\sqrt{x^2+1}}$ | $\dfrac{x}{\sqrt{1-x^2}}$ |
| Domain: | $(-\infty,\infty)$ | $(-\infty,\infty)$ | $(-1,1)$ |
| Range: | $[0,1)$ | $(-1,1)$ | $(-\infty,\infty)$ |
| Graph: |  |
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| Compositions: | $\begin{array}{ll} \sin(\operatorname{ arcsec}_1 x) \\ \cos(\operatorname{ arccsc}_1 x) \end{array}$ | $\begin{array}{ll} \sin(\operatorname{ arcsec}_2 x) \\ \cos(\operatorname{ arccsc}_2 x) \end{array}$ | $\begin{array}{ll} \sec(\arcsin x) \\ \csc(\arccos x) \end{array}$ |
| Simplified: | $\dfrac{\sqrt{x^2-1}}{|x|}$ | $\dfrac{\sqrt{x^2-1}}{x}$ | $\dfrac{1}{\sqrt{1-x^2}}$ |
| Domain: | $(-\infty,1] \cup [1,\infty)$ | $(-\infty,1] \cup [1,\infty)$ | $(-1,1)$ |
| Range: | $[0,1)$ | $(-1,1)$ | $[1,\infty)$ |
| Graph: |  |
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| Compositions: | $\sin( \operatorname{ arccot}_1 x)$ | $\begin{array}{ll} \cos(\arctan x) \\ \sin( \operatorname{ arccot}_2 x) \end{array}$ | $\begin{array}{ll} \tan(\arccos x) \\ \cot(\arcsin x) \end{array}$ |
| Simplified: | $\dfrac{\pm 1}{\sqrt{x^2+1}}$ | $\dfrac{1}{\sqrt{x^2+1}}$ | $\dfrac{\sqrt{1-x^2}}{x}$ |
| Domain: | $(-\infty,\infty)$ | $(-\infty,\infty)$ | $[-1,0) \cup (0,1]$ |
| Range: | $(-1,0) \cup (0,1]$ | $(0,1]$ | $(-\infty,\infty)$ |
| Graph: |  |
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Now we turn to the 54 compositions where the inverse trigonometric function is the outside function. The first 18 such compositions result in 18 different graphs, each of which is piecewise linear. Therefore, each of these compositions will have an algebraic simplification for each linear piece of the graph. We provide the simplification only for that piece corresponding to the first quadrant of the unit circle.
| Compositions: | $\arcsin(\sin x)$ | $\arccos(\cos x)$ | $\arctan(\tan x)$ |
| Simplified, on $\left(0,\dfrac{\pi}{2}\right)$: | $x$ | $x$ | $x$ |
| Domain: | $(-\infty,\infty)$ | $(-\infty,\infty)$ | $\bigcup\limits_{k=-\infty}^\infty \left(\left(k-\dfrac12\right)\pi,\left(k+\dfrac12\right)\pi\right)$ |
| Range: | $\left[ -\dfrac{\pi}{2},\dfrac{\pi}{2}\right]$ | $[0,\pi]$ | $\left( -\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$ |
| Graph: |  |
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| Compositions: | $\operatorname{ arccot}_1(\cot x)$ | $\operatorname{ arccot}_2(\cot x)$ | $\operatorname{ arcsec}_1(\sec x)$ |
| Simplified, on $\left(0,\dfrac{\pi}{2}\right)$: | $x$ | $x$ | $x$ |
| Domain: | $\bigcup\limits_{ k=-\infty}^\infty \left(k\pi,(k+1)\pi \right)$ | $\bigcup\limits_{ k=-\infty}^\infty \left(k\pi,(k+1)\pi \right)$ | $\bigcup\limits_{ k=-\infty}^\infty \left(\left( k-\dfrac12\right)\pi, \left(k+\dfrac12\right) \pi\right)$ |
| Range: | $\left(-\dfrac{\pi}{2},0\right) \cup \left(0,\dfrac{\pi}{2}\right]$ | $(0,\pi)$ | $\left[0,\dfrac{\pi}{2}\right) \cup \left(\dfrac{\pi}{2},\pi \right]$ |
| Graph: |  |
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| Compositions: | $\operatorname{ arcsec}_2(\sec x)$ | $\operatorname{ arccsc}_1(\csc x)$ | $\operatorname{ arccsc}_2(\csc x)$ |
| Simplified, on $\left(0,\dfrac{\pi}{2}\right)$: | $x$ | $x$ | $x$ |
| Domain: | $\bigcup\limits_{ k=-\infty}^\infty \left(\left( k-\dfrac12\right)\pi, \left(k+\dfrac12\right) \pi\right)$ | $\bigcup\limits_{ k=-\infty}^\infty \left(k\pi,(k+1)\pi \right)$ | $\bigcup\limits_{ k=-\infty}^\infty \left(k\pi,(k+1)\pi \right)$ |
| Range: | $\left[-\pi,-\dfrac{\pi}{2}\right) \cup \left[0,\dfrac{\pi}{2}\right)$ | $\left[-\dfrac{\pi}{2},0\right) \cup \left(0,\dfrac{\pi}{2}\right]$ | $\left(-\pi,-\dfrac{\pi}{2}\right] \cup \left(0,\dfrac{\pi}{2}\right]$ |
| Graph: |  |
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| Compositions: | $\arcsin(\cos x)$ | $\arccos(\sin x)$ | $\arctan(\cot x)$ |
| Simplified, on $\left(0,\dfrac{\pi}{2}\right)$: | $\dfrac{\pi}{2}-x$ | $\dfrac{\pi}{2}-x$ | $\dfrac{\pi}{2}-x$ |
| Domain: | $(-\infty,\infty)$ | $(-\infty,\infty)$ | $\bigcup\limits_{ k=-\infty}^\infty \left(k\pi,(k+1)\pi \right)$ |
| Range: | $\left[ -\dfrac{\pi}{2},\dfrac{\pi}{2}\right]$ | $[0,\pi]$ | $\left( -\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$ |
| Graph: |  |
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| Compositions: | $\operatorname{ arccot}_1(\tan x)$ | $\operatorname{ arccot}_2(\tan x)$ | $\operatorname{ arcsec}_1(\csc x)$ |
| Simplified, on $\left(0,\dfrac{\pi}{2}\right)$: | $\dfrac{\pi}{2}-x$ | $\dfrac{\pi}{2}-x$ | $\dfrac{\pi}{2}-x$ |
| Domain: | $\bigcup\limits_{ k=-\infty}^\infty \left(\left( k-\dfrac12\right)\pi, \left(k+\dfrac12\right) \pi\right)$ | $\bigcup\limits_{ k=-\infty}^\infty \left(\left( k-\dfrac12\right)\pi, \left(k+\dfrac12\right) \pi\right)$ | $\bigcup\limits_{ k=-\infty}^\infty \left(k\pi,(k+1)\pi \right)$ |
| Range: | $\left( -\dfrac{\pi}{2},0\right) \cup \left(0, \dfrac{\pi}{2}\right]$ | $(0,\pi)$ | $\left[0, \dfrac{\pi}{2},0\right) \cup \left(\dfrac{\pi}{2},\pi\right]$ |
| Graph: |  |
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| Compositions: | $\operatorname{ arcsec}_2(\csc x)$ | $\operatorname{ arccsc}_1(\sec x)$ | $\operatorname{ arccsc}_2(\sec x)$ |
| Simplified, on $\left(0,\dfrac{\pi}{2}\right)$: | $\dfrac{\pi}{2}-x$ | $\dfrac{\pi}{2}-x$ | $\dfrac{\pi}{2}-x$ |
| Domain: | $\bigcup\limits_{ k=-\infty}^\infty \left(k\pi,(k+1)\pi \right)$ | $\bigcup\limits_{ k=-\infty}^\infty \left(\left( k-\dfrac12\right)\pi, \left(k+\dfrac12\right) \pi\right)$ | $\bigcup\limits_{ k=-\infty}^\infty \left(\left( k-\dfrac12\right)\pi, \left(k+\dfrac12\right) \pi\right)$ |
| Range: | $\left[-\pi, -\dfrac{\pi}{2}\right) \cup \left[0, \dfrac{\pi}{2}\right)$ | $\left[ -\dfrac{\pi}{2},0\right) \cup \left(0, \dfrac{\pi}{2}\right]$ | $\left(-\pi, -\dfrac{\pi}{2}\right] \cup \left(0, \dfrac{\pi}{2}\right]$ |
| Graph: |  |
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The next 12 compositions give 12 different graphs which have a wave-like appearance. However, these compositions do not simplify algebraically (or trigonometrically). In fact, the Fourier expansion of the first of these compositions has an infinite number of sinusoidal terms, and is specifically:
\begin{equation*} \arctan(\sin x)=\sum\limits_{n=1}^\infty \dfrac{2\left(-1+\sqrt{2}\right)^{2n-1}}{2n-1} \sin(2n-1)x \end{equation*}| Compositions: | $\arctan (\sin x)$ | $\arctan (\cos x)$ | $\operatorname{ arccot}_2 (\sin x)$ |
| Domain: | $(-\infty,\infty)$ | $(-\infty,\infty)$ | $(-\infty,\infty)$ |
| Range: | $\left[ -\dfrac{\pi}{4}, \dfrac{\pi}{4} \right]$ | $\left[ -\dfrac{\pi}{4}, \dfrac{\pi}{4} \right]$ | $\left[ \dfrac{\pi}{4}, \dfrac{3\pi}{4} \right]$ |
| Graph: |  |
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| Compositions: | $\operatorname{ arccot}_2 (\cos x)$ | $\operatorname{ arccot}_1 (\sec x)$ | $\operatorname{ arccot}_1 (\csc x)$ |
| Domain: | $(-\infty,\infty)$ | $\bigcup\limits_{ k=-\infty}^\infty \left(\left( k-\dfrac12\right)\pi, \left(k+\dfrac12\right) \pi\right)$ | $\bigcup\limits_{ k=-\infty}^\infty \left(k\pi,(k+1)\pi \right)$ |
| Range: | $\left[ \dfrac{\pi}{4}, \dfrac{3\pi}{4} \right]$ | $\left[ -\dfrac{\pi}{4},0\right) \cup \left(0, \dfrac{\pi}{4} \right]$ | $\left[ -\dfrac{\pi}{4},0\right) \cup \left(0, \dfrac{\pi}{4} \right]$ |
| Graph: |  |
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| Compositions: | $\operatorname{ arccot}_1 (\sin x)$ | $\operatorname{ arccot}_1 (\cos x)$ | $\arctan (\sec x)$ |
| Domain: | $(-\infty,\infty)$ | $(-\infty,\infty)$ | $\bigcup\limits_{ k=-\infty}^\infty \left(\left( k-\dfrac12\right)\pi, \left(k+\dfrac12\right) \pi\right)$ |
| Range: | $\left(-\dfrac{\pi}{2}, -\dfrac{\pi}{4} \right] \cup \left[\dfrac{\pi}{4}, \dfrac{\pi}{2} \right]$ | $\left(-\dfrac{\pi}{2}, -\dfrac{\pi}{4} \right] \cup \left[\dfrac{\pi}{4}, \dfrac{\pi}{2} \right]$ | $\left(-\dfrac{\pi}{2}, -\dfrac{\pi}{4} \right] \cup \left[\dfrac{\pi}{4}, \dfrac{\pi}{2} \right)$ |
| Graph: |  |
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| Compositions: | $\arctan (\csc x)$ | $\operatorname{ arccot}_2 (\sec x)$ | $\operatorname{ arccot}_2 (\csc x)$ |
| Domain: | $\bigcup\limits_{ k=-\infty}^\infty \left(k\pi,(k+1)\pi \right)$ | $\bigcup\limits_{ k=-\infty}^\infty \left(\left( k-\dfrac12\right)\pi, \left(k+\dfrac12\right) \pi\right)$ | $\bigcup\limits_{ k=-\infty}^\infty \left(k\pi,(k+1)\pi \right)$ |
| Range: | $\left(-\dfrac{\pi}{2}, -\dfrac{\pi}{4} \right] \cup \left[\dfrac{\pi}{4}, \dfrac{\pi}{2} \right)$ | $\left(0, \dfrac{\pi}{4} \right] \cup \left[\dfrac{3\pi}{4}, \pi\right)$ | $\left(0, \dfrac{\pi}{4} \right] \cup \left[\dfrac{3\pi}{4}, \pi\right)$ |
| Graph: |  |
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The next 12 compositions give 12 periodic graphs where each branch has an appearance similar to (but not exactly the same as) a basic arcsine graph.
| Compositions: | $\arcsin (\tan x)$ | $\arccos (\tan x)$ |
| Domain: | $\bigcup\limits_{k=-\infty}^\infty \left[\left(k-\dfrac14\right)\pi, \left(k+ \dfrac14\right)\pi\right]$ | |
| Range: | $\left[ -\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$ | $[0, \pi]$ |
| Graph: |  |
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| Compositions: | $\arcsin (\cot x)$ | $\arccos (\cot x)$ |
| Domain: | $\bigcup\limits_{k=-\infty}^\infty \left[\left(k+\dfrac14\right)\pi, \left(k+ \dfrac34\right)\pi\right]$ | |
| Range: | $\left[ -\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$ | $[0, \pi]$ |
| Graph: |  |
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| Compositions: | $\operatorname{ arcsec}_1 (\tan x)$ | $\operatorname{ arccsc}_1 (\tan x)$ |
| Domain: | $\bigcup\limits_{k=-\infty}^\infty \left[\left(k+\dfrac14\right)\pi, \left(k+\dfrac12\right)\pi\right) \cup \left(\left(k+\dfrac12\right)\pi, \left(k+\dfrac34\right)\pi\right]$ | |
| Range: | $\left[0, \dfrac{\pi}{2}\right) \cup \left(\dfrac{\pi}{2},\pi\right]$ | $\left[-\dfrac{\pi}{2},0\right) \cup \left(0,\dfrac{\pi}{2}\right]$ |
| Graph: |  |
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| Compositions: | $\operatorname{ arcsec}_1 (\cot x)$ | $\operatorname{ arccsc}_1 (\cot x)$ |
| Domain: | $\bigcup\limits_{k=-\infty}^\infty \left[\left(k-\dfrac14\right)\pi, k\pi\right) \cup \left(k\pi, \left(k+\dfrac14\right)\pi\right]$ | |
| Range: | $\left[0, \dfrac{\pi}{2}\right) \cup \left(\dfrac{\pi}{2},\pi\right]$ | $\left[-\dfrac{\pi}{2},0\right) \cup \left(0,\dfrac{\pi}{2}\right]$ |
| Graph: |  |
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| Compositions: | $\operatorname{ arcsec}_2 (\tan x)$ | $\operatorname{ arccsc}_2 (\tan x)$ |
| Domain: | $\bigcup\limits_{k=-\infty}^\infty \left[\left(k+\dfrac14\right)\pi, \left(k+\dfrac12\right)\pi\right) \cup \left(\left(k+\dfrac12\right)\pi, \left(k+\dfrac34\right)\pi\right]$ | |
| Range: | $\left[-\pi, -\dfrac{\pi}{2}\right) \cup \left[0,\dfrac{\pi}{2}\right)$ | $\left(-\pi, -\dfrac{\pi}{2}\right] \cup \left(0,\dfrac{\pi}{2}\right]$ |
| Graph: |  |
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| Compositions: | $\operatorname{ arcsec}_2 (\cot x)$ | $\operatorname{ arccsc}_2 (\cot x)$ |
| Domain: | $\bigcup\limits_{k=-\infty}^\infty \left[\left(k-\dfrac14\right)\pi, k\pi\right) \cup \left(k\pi, \left(k+\dfrac14\right)\pi\right]$ | |
| Range: | $\left[-\pi, -\dfrac{\pi}{2}\right) \cup \left[0,\dfrac{\pi}{2}\right)$ | $\left(-\pi, -\dfrac{\pi}{2}\right] \cup \left(0,\dfrac{\pi}{2}\right]$ |
| Graph: |  |
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The last 12 compositions are undefined almost everywhere (except on a countable subset of real numbers), and each of the 6 graphs which result consist of a periodic sequence of isolated points.
| Compositions: | $\begin{array}{ll} \arccos(\sec x) \\ \operatorname{ arcsec}_1 (\cos x) \end{array}$ | $\begin{array}{ll} \arccos(\csc x) \\ \operatorname{ arcsec}_1 (\sin x) \end{array}$ |
| Domain: | $\bigcup\limits_{k=-\infty}^\infty \{ k\pi \}$ | $\bigcup\limits_{k=-\infty}^\infty \left\{ \left(k+\dfrac12\right)\pi \right\}$ |
| Range: | $\{0,\pi\}$ | $\{0,\pi\}$ |
| Graph: |  |
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| Compositions: | $\begin{array}{ll} \arcsin(\sec x) \\ \operatorname{ arccsc}_1 (\cos x) \\ \operatorname{ arccsc}_2 (\cos x) \end{array}$ | $\begin{array}{ll} \arcsin(\csc x) \\ \operatorname{ arccsc}_1 (\sin x) \\ \operatorname{ arccsc}_2 (\sin x) \end{array}$ |
| Domain: | $\bigcup\limits_{k=-\infty}^\infty \{ k\pi \}$ | $\bigcup\limits_{k=-\infty}^\infty \left\{ \left(k+\dfrac12\right)\pi \right\}$ |
| Range: | $\left\{ -\dfrac{\pi}{2}, \dfrac{\pi}{2} \right\}$ | $\left\{ -\dfrac{\pi}{2}, \dfrac{\pi}{2} \right\}$ |
| Graph: |  |
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| Compositions: | $\operatorname{ arcsec}_2 (\cos x)$ | $\operatorname{ arcsec}_2 (\sin x)$ |
| Domain: | $\bigcup\limits_{k=-\infty}^\infty \{ k\pi \}$ | $\bigcup\limits_{k=-\infty}^\infty \left\{ \left(k+\dfrac12\right)\pi \right\}$ |
| Range: | $\{-\pi,0\}$ | $\{-\pi,0\}$ |
| Graph: |  |
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