Basic Inverse Trig Function Identities

The basic inverse trigonometric identities come in several varieties. These include reciprocal, symmetric, and cofunction identities.

Inverse Reciprocal Identities Theorem

The following identities are true for all values for which they are defined:

 $\arcsin x=\operatorname{ arccsc}_1 \dfrac{1}{x} = \left\{ \begin{array}{ll} \operatorname{ arccsc}_2 \dfrac{1}{x} & \text{if } 0< x\le 1 \\ -\pi-\operatorname{ arccsc}_2 \dfrac{1}{x} & \text{if } -1\le x < 0 \end{array} \right.$ $\arccos x=\operatorname{ arcsec}_1 \dfrac{1}{x} = \left\{ \begin{array}{ll} \operatorname{arcsec}_2 \dfrac{1}{x} & \text{if } 0< x\le 1 \\ 2\pi-\operatorname{ arcsec}_2 \dfrac{1}{x} & \text{if } -1\le x < 0 \end{array} \right.$ $\arctan x=\operatorname{ arccot}_2 \dfrac{1}{x} = \left\{ \begin{array}{ll} \operatorname{arccot}_1 \dfrac{1}{x} & \text{if } x>0 \\ \pi+\operatorname{ arccot}_1 \dfrac{1}{x} & \text{if } x < 0 \end{array} \right.$ $\operatorname{ arccot}_1 x = \left\{ \begin{array}{ll} \arctan \dfrac{1}{x} & \text{if } x>0 \\ -\pi+\arctan \dfrac{1}{x} & \text{if } x < 0 \end{array} \right.$ $\operatorname{ arccot}_2 x = \arctan\dfrac{1}{x}$ $\operatorname{ arcsec}_1 x = \arccos\dfrac{1}{x}$ $\operatorname{ arcsec}_2 x = \left\{ \begin{array}{ll} \arccos \dfrac{1}{x} & \text{if } x\ge 1 \\ 2\pi-\arccos \dfrac{1}{x} & \text{if } x \le -1 \end{array} \right.$ $\operatorname{ arccsc}_1 x = \arcsin\dfrac{1}{x}$ $\operatorname{ arccsc}_2 x = \left\{ \begin{array}{ll} \arcsin \dfrac{1}{x} & \text{if } x\ge 1 \\ -\pi-\arcsin \dfrac{1}{x} & \text{if } x \le -1 \end{array} \right.$

Proof: The proof of the first equality uses the inverse trig definitions and the Reciprocal Identities Theorem. We first note that the ranges of the inverse sine function and the first inverse cosecant function are almost identical, then proceed as follows:

\begin{align} y &= \arcsin x \\ x &= \sin y \\ \dfrac{1}{x} &= \csc y \\ \operatorname{ arccsc}_1 \dfrac{1}{x} &= y \\ \operatorname{ arccsc}_1 \dfrac{1}{x} &= \arcsin x \end{align}

The proofs of the other identities are similar, but extreme care must be taken with the intervals of domain and range on which the definitions are valid.♦

Inverse Symmetric Identities Theorem

The following identities are true for all values for which they are defined:

 $\arcsin(-x)=-\arcsin x$ $\arccos(-x)=\pi-\arccos x$ $\arctan(-x)=-\arctan x$ $\operatorname{ arccot}_1 (-x)=\pi-\operatorname{ arccot}_1 x$ $\operatorname{ arccot}_2 (-x)=-\operatorname{ arccot}_2 x$ $\operatorname{ arcsec}_1 (-x)=\pi-\operatorname{ arcsec}_1 x$ $\operatorname{ arcsec}_2 (-x)=\left\{ \begin{array}{ll} \pi+\operatorname{ arcsec}_2 x & \text{if } x\ge 1 \\ -\pi+\operatorname{ arcsec}_2 x & \text{if } x\le 1 \end{array} \right.$ $\operatorname{ arccsc}_1 (-x)=-\operatorname{ arccsc}_1 x$ $\operatorname{ arccsc}_2 (-x)=\left\{ \begin{array}{ll} -\pi+\operatorname{ arccsc}_2 x & \text{if } x\ge 1 \\ \pi+\operatorname{ arccsc}_2 x & \text{if } x\le 1 \end{array} \right.$

Proof: The proof of the first identity involves the symmetric identity for the sine function.

\begin{align} y &= \arcsin(-x) \\ -x &= \sin y \\ x &= -\sin y \\ x &= \sin(-y) \\ -y &= \arcsin x \\ y &= -\arcsin x \\ \arcsin(-x) &= -\arcsin x \end{align}

The proofs of the other identities are similar, but once again care must be taken to identify the correct ranges of each function.♦

Inverse Cofunction Identities Theorem

The following identities are true for all values for which they are defined:

 $\arcsin x + \arccos x = \dfrac{\pi}{2}$ $\arctan x + \operatorname{ arccot}_1 x = \dfrac{\pi}{2}$ $\arctan x + \operatorname{ arccot}_2 x = \left\{ \begin{array}{ll} \dfrac{\pi}{2} & \text{if } x\ge 0 \\ -\dfrac{\pi}{2} & \text{if } x<0 \end{array} \right.$ $\operatorname{ arcsec}_1 x+ \operatorname{ arccsc}_1 x=\dfrac{\pi}{2}$ $\operatorname{ arcsec}_2 x+ \operatorname{ arccsc}_2 x=\dfrac{\pi}{2}$

Proof: The proof of the Inverse Cofunction Identities Theorem uses the Cofunction Theorem. The proof of the first equality proceeds as follows:

\begin{align} y &= \arcsin x \\ \sin y &= x \\ \cos \left(\dfrac{\pi}{2}-y\right) &= x \\ \dfrac{\pi}{2}-y &= \arccos x \\ \dfrac{\pi}{2}-\arcsin x &= \arccos x \\ \arcsin x + \arccos x &= \dfrac{\pi}{2} \end{align}

If the two versions of the arccotangent function are compared, it will be noted that version 1 is continuous and provides the nicest form for the inverse cofunction identity, but version 2 provides the nicest forms for the inverse reciprocal and inverse symmetric identities.