Properties of Inverse Trigonometric Functions

Table Of Contents

• Properties of Inverse Trigonometric Functionss
• Inverse of Negative x
• Reciprocal of x
• Addition of Different Inverse function
• Addition of Same Inverse function
• Double angle identity for Inverse functions
• Odd and Even Properties

Properties of Inverse Trigonometric Functions

• In last page,we cover about what are Inverse Trigonometric Functions and how they are used to find the angles corresponding to given trigonometric ratios, which is an essential concept in mathematics, physics, and engineering
• In this article, we will discuss the properties of these inverse trigonometric functions.
• It may be mentioned here that these results are valid within the principal value branches of the corresponding inverse trigonometric functions and wherever they are defined.
• Some results may not be valid for all values of the domains of inverse trigonometric functions

Inverse of Negative x

$sin^{-1} (-x) = -sin^{-1} (x)$
$cos^{-1} (-x) = \pi - cos^{-1} (x)$
$tan^{-1} (-x) = -tan^{-1} (x)$
$sec^{-1} (-x) = \pi - sec^{-1} (x)$
$cosec^{-1} (-x) = -cosec^{-1} (x)$
$cot^{-1} (-x) = \pi - cot^{-1} (x)$
Lets see the proof of one of them
Proof on Sin
Let $sin^{-1} (-x) = y$
Then $-x = sin y$
or $x = - sin y$
or $x = sin (-y)$
Hence $sin^{-1}x = -y = - sin^{-1} (-x)$
Similary we can prove for others

Reciprocal of x

$sin^{-1} (\frac {1}{x}) = cosec^{-1} (x)$
$cos^{-1} (\frac {1}{x}) = sec^{-1} (x)$
$tan^{-1} (\frac {1}{x}) = cot^{-1} (x)$

Complementary Inverse function

$sin^{-1} (x) + cos^ {-1} (x) = \frac {\pi}{2}$
$sec^{-1} (x) + cosec^ {-1} (x) = \frac {\pi}{2}$
$tan^{-1} (x) + cot^ {-1} (x) = \frac {\pi}{2}$

Addition of Same Inverse function

$sin^{-1} (x) + sin^ {-1} (y) = sin ^{-1} (x \sqrt {1-y^2} + y \sqrt {1-x^2})$ if $x,y \geq 0 $, $x^2 + y^2 \leq 1$
$sin^{-1} (x) + sin^ {-1} (y) = \pi - sin ^{-1} (x \sqrt {1-y^2} + y \sqrt {1-x^2})$ if $x,y \geq 0 $, $x^2 + y^2 > 1$
$sin^{-1} (x) - sin^ {-1} (y) = sin ^{-1} (x \sqrt {1-y^2} - y \sqrt {1-x^2)}$ if $x,y \geq 0 $, $x^2 + y^2 \leq 1$
$sin^{-1} (x) - sin^ {-1} (y) = \pi - sin ^{-1} (x \sqrt {1-y^2} - y \sqrt {1-x^2})$ if $x,y \geq 0 $, $x^2 + y^2 > 1$
$cos^{-1} (x) + cos^ {-1} (y) = cos ^{-1} (x y - \sqrt {1-y^2} \sqrt {1-x^2})$ if $x,y \geq 0 $, $x^2 + y^2 \leq 1$
$cos^{-1} (x) + cos^ {-1} (y) = \pi - cos ^{-1} ((x y - \sqrt {1-y^2} \sqrt {1-x^2})$ if $x,y \geq 0 $, $x^2 + y^2 > 1$
$cos^{-1} (x) - cos^ {-1} (y) = cos ^{-1} (x y + \sqrt {1-y^2} \sqrt {1-x^2})$ if $x,y \geq 0 $, $x^2 + y^2 \leq 1$
$cos^{-1} (x) - cos^ {-1} (y) = \pi - cos ^{-1} (x y + \sqrt {1-y^2} \sqrt {1-x^2})$ if $x,y \geq 0 $, $x^2 + y^2 > 1$
$tan^{-1} (x) + tan^ {-1} (y)= tan^{-1} (\frac {x+y}{1-xy})$ , if $x,y > 0 $, $xy < 1$
$tan^{-1} (x) + tan^ {-1} (y)= \pi + tan^{-1} (\frac {x+y}{1-xy})$ , if $x,y > 0 $, $xy > 1$
$tan^{-1} (x) + tan^ {-1} (y)= tan^{-1} (\frac {x+y}{1-xy}) - \pi$ , if $x < 0, y > 0 $, $xy > 1$
$tan^{-1} (x) - tan^ {-1} (y)= tan^{-1} (\frac {x-y}{1+xy}) - \pi$ , if $xy > -1$
$tan^{-1} (x) + tan^ {-1} (y) + tan^ {-1} (z) = tan^{-1} (\frac {x+y+z - xyz}{1-xy-yz-xz})$

Double angle identity for Inverse functions

$ 2 sin^{-1} (x) = sin^{-1} (2x \sqrt {1-x^2})$ if $ -\frac {1}{\sqrt {2}} \leq x \frac {1}{\sqrt {2}} $
$ 2 cos^{-1} (x) = cos^{-1} (2x^2 -1)$
$2 tan^{-1} (x) = tan^{-1} (\frac {2x}{1-x^2})$ if $ -1 $2 tan^{-1} (x) = sin^{-1} (\frac {2x}{1+x^2})$ if $ |x| \leq 1$
$2 tan^{-1} (x) = cos^{-1} (\frac {1 -x^2}{1+x^2})$ if $ x \geq 0$
$3 sin^{-1} (x) = sin^{-1} (3x -4x^3)$
$3 cos^{-1} (x) = cos^{-1} (4x^3 - 3x)$
$3 tan^{-1} (x) = tan^{-1} (\frac {3x -x^3}{1-3x^2})$

Odd and Even Properties

Arcsine and arctangent are odd functions
$sin^{-1}(-x) = -sin^{-1} (x)$
$tan^{-1}(-x) = -tan^{-1}x$

Also Read

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• Notes
  • Inverse Trigonometric Functions
  • Properties of Inverse Trigonometric Functions
  NCERT Solutions & Assignments
  • NCERT Solution for Inverse Trigonometric Functions Exercise 2.1
  • NCERT Solution for Inverse Trigonometric Functions Exercise 2.2
  • Inverse Trigonometric Functions Extra Questions for Class 12

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