JEE Mathematics Inverse Trigonometric Functions

Inverse Trigonometric Functions Formulas

Principal Values & Domains of Inverse Trigonometric/Circular Functions:
(i) \( y = \sin^{-1} x \) where Domain \( -1 \le x \le 1 \), Range \( -\frac{\pi}{2} \le y \le \frac{\pi}{2} \)
(ii) \( y = \cos^{-1} x \) where Domain \( -1 \le x \le 1 \), Range \( 0 \le y \le \pi \)
(iii) \( y = \tan^{-1} x \) where Domain \( x \in R \), Range \( -\frac{\pi}{2} < y < \frac{\pi}{2} \)
(iv) \( y = \text{cosec}^{-1} x \) where Domain \( x \le -1 \text{ or } x \ge 1 \), Range \( -\frac{\pi}{2} \le y \le \frac{\pi}{2}, y \neq 0 \)
(v) \( y = \sec^{-1} x \) where Domain \( x \le -1 \text{ or } x \ge 1 \), Range \( 0 \le y \le \pi; y \neq \frac{\pi}{2} \)
(vi) \( y = \cot^{-1} x \) where Domain \( x \in R \), Range \( 0 < y < \pi \)

\( \tan^{-1} (\tan x) = x; -\frac{\pi}{2} < x < \frac{\pi}{2} \)
\( \tan^{-1} (\tan x) = x; -\frac{\pi}{2} < x < \frac{\pi}{2} \)

\( \cot^{-1} (\cot x) = x; 0 < x < \pi \)
\( \cot^{-1} (\cot x) = x; 0 < x < \pi \)

\( \sec^{-1} (\sec x) = x; 0 \le x \le \pi, x \neq \frac{\pi}{2} \)
\( \sec^{-1} (\sec x) = x; 0 \le x \le \pi, x \neq \frac{\pi}{2} \)

\( \text{cosec}^{-1} (\text{cosec } x) = x; x \neq 0, -\frac{\pi}{2} \le x \le \frac{\pi}{2} \)
\( \text{cosec}^{-1} (\text{cosec } x) = x; x \neq 0, -\frac{\pi}{2} \le x \le \frac{\pi}{2} \)

\( \sin^{-1} (-x) = -\sin^{-1} x, -1 \le x \le 1 \)
\( \sin^{-1} (-x) = -\sin^{-1} x, -1 \le x \le 1 \)

\( \tan^{-1} (-x) = -\tan^{-1} x, x \in R \)
\( \tan^{-1} (-x) = -\tan^{-1} x, x \in R \)

\( \cos^{-1} (-x) = \pi - \cos^{-1} x, -1 \le x \le 1 \)
\( \cos^{-1} (-x) = \pi - \cos^{-1} x, -1 \le x \le 1 \)

\( \cot^{-1} (-x) = \pi - \cot^{-1} x, x \in R \)
\( \cot^{-1} (-x) = \pi - \cot^{-1} x, x \in R \)

\( \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}, -1 \le x \le 1 \)
\( \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}, -1 \le x \le 1 \)

\( \tan^{-1} x + \cot^{-1} x = \frac{\pi}{2}, x \in R \)
\( \tan^{-1} x + \cot^{-1} x = \frac{\pi}{2}, x \in R \)

\( \text{cosec}^{-1} x + \sec^{-1} x = \frac{\pi}{2}, |x| \ge 1 \)
\( \text{cosec}^{-1} x + \sec^{-1} x = \frac{\pi}{2}, |x| \ge 1 \)

Identities of Addition and Subtraction Formulas

\( \sin^{-1} x + \sin^{-1} y \)
\( \sin^{-1} \left[ x\sqrt{1-y^2} + y\sqrt{1-x^2} \right], x \ge 0, y \ge 0 \text{ & } (x^2 + y^2) \le 1 \)
\( = \pi - \sin^{-1} \left[ x\sqrt{1-y^2} + y\sqrt{1-x^2} \right], x \ge 0, y \ge 0 \text{ & } x^2 + y^2 > 1 \)

\( \cos^{-1} x + \cos^{-1} y \)
\( \cos^{-1} \left[ xy - \sqrt{1-x^2}\sqrt{1-y^2} \right], x \ge 0, y \ge 0 \)

\( \tan^{-1} x + \tan^{-1} y \)
\( \tan^{-1} \frac{x+y}{1-xy}, x > 0, y > 0 \text{ & } xy < 1 \)
\( = \pi + \tan^{-1} \frac{x+y}{1-xy}, x > 0, y > 0 \text{ & } xy > 1 = \frac{\pi}{2}, x > 0, y > 0 \text{ & } xy = 1 \)

\( \sin^{-1} x - \sin^{-1} y \)
\( \sin^{-1} \left[ x\sqrt{1-y^2} - y\sqrt{1-x^2} \right], x \ge 0, y \ge 0 \)

\( \cos^{-1} x - \cos^{-1} y \)
\( \cos^{-1} \left[ xy + \sqrt{1-x^2}\sqrt{1-y^2} \right], x \ge 0, y \ge 0, x \le y \)

\( \tan^{-1} x - \tan^{-1} y \)
\( \tan^{-1} \frac{x-y}{1+xy}, x > 0, y > 0 \)

\( \sin^{-1} \left( 2x\sqrt{1-x^2} \right) \)
\( \begin{cases} 2 \sin^{-1} x & \text{if } |x| \le \frac{1}{\sqrt{2}} \\ \pi - 2 \sin^{-1} x & \text{if } x > \frac{1}{\sqrt{2}} \\ -(\pi + 2 \sin^{-1} x) & \text{if } x < -\frac{1}{\sqrt{2}} \end{cases} \)

\( \cos^{-1} (2x^2 - 1) \)
\( \begin{cases} 2 \cos^{-1} x & \text{if } 0 \le x \le 1 \\ 2\pi - 2 \cos^{-1} x & \text{if } -1 \le x < 0 \end{cases} \)

\( \tan^{-1} \frac{2x}{1-x^2} \)
\( \begin{cases} 2 \tan^{-1} x & \text{if } |x| < 1 \\ \pi + 2 \tan^{-1} x & \text{if } x < -1 \\ -(\pi - 2 \tan^{-1} x) & \text{if } x > 1 \end{cases} \)

\( \sin^{-1} \frac{2x}{1+x^2} \)
\( \begin{cases} 2 \tan^{-1} x & \text{if } |x| \le 1 \\ \pi - 2 \tan^{-1} x & \text{if } x > 1 \\ -(\pi + 2 \tan^{-1} x) & \text{if } x < -1 \end{cases} \)

\( \cos^{-1} \frac{1-x^2}{1+x^2} \)
\( \begin{cases} 2 \tan^{-1} x & \text{if } x \ge 0 \\ -2 \tan^{-1} x & \text{if } x < 0 \end{cases} \)

\( \tan^{-1} x + \tan^{-1} y + \tan^{-1} z \)
\( \tan^{-1} \left[ \frac{x + y + z - xyz}{1 - xy - yz - zx} \right] \text{ if, } x > 0, y > 0, z > 0 \text{ & } (xy + yz + zx) < 1 \)

• If \( \tan^{-1} x + \tan^{-1} y + \tan^{-1} z = \pi \) then \( x + y + z = xyz \)
• If \( \tan^{-1} x + \tan^{-1} y + \tan^{-1} z = \frac{\pi}{2} \) then \( xy + yz + zx = 1 \)
• \( \tan^{-1} 1 + \tan^{-1} 2 + \tan^{-1} 3 = \pi \)
• \( \tan^{-1} 1 + \tan^{-1} \frac{1}{2} + \tan^{-1} \frac{1}{3} = \frac{\pi}{2} \)