A
Negation Properties (Odd & Even)
Effect of replacing $x$ with $-x$ in each inverse function
Key Pattern
$\sin^{-1}$, $\tan^{-1}$, $\csc^{-1}$ are odd functions (negation gives $-f(x)$).
$\cos^{-1}$, $\sec^{-1}$, $\cot^{-1}$ are neither odd nor even (negation gives $\pi - f(x)$).
sin⁻¹
$\sin^{-1}(-x) = -\sin^{-1} x$
▸ Odd function
cos⁻¹
$\cos^{-1}(-x) = \pi - \cos^{-1} x$
▸ Neither odd nor even
tan⁻¹
$\tan^{-1}(-x) = -\tan^{-1} x$
▸ Odd function
cot⁻¹
$\cot^{-1}(-x) = \pi - \cot^{-1} x$
▸ Neither odd nor even
sec⁻¹
$\sec^{-1}(-x) = \pi - \sec^{-1} x$
▸ Neither odd nor even
csc⁻¹
$\csc^{-1}(-x) = -\csc^{-1} x$
▸ Odd function
B
Complementary Pairs (Sum = π/2)
Each function and its co-function always sum to π/2
$\sin^{-1} x$
$+ \cos^{-1} x$
$+ \cos^{-1} x$
= π/2
$x \in [-1,\,1]$
$\tan^{-1} x$
$+ \cot^{-1} x$
$+ \cot^{-1} x$
= π/2
$x \in \mathbb{R}$
$\sec^{-1} x$
$+ \csc^{-1} x$
$+ \csc^{-1} x$
= π/2
$|x| \geq 1$
Trick: These follow directly from the co-function identity: e.g., $\sin\theta = \cos(\pi/2 - \theta)$, so $\sin^{-1}x + \cos^{-1}x = \pi/2$.
C
Addition Formulae for tan⁻¹
Sum and difference of two inverse tangent values
Addition — three cases
$$\tan^{-1} x + \tan^{-1} y = \begin{cases} \tan^{-1}\!\left(\dfrac{x+y}{1-xy}\right) & \text{if } xy < 1 \\[10pt] \pi + \tan^{-1}\!\left(\dfrac{x+y}{1-xy}\right) & \text{if } xy > 1,\; x > 0,\; y > 0 \\[10pt] -\pi + \tan^{-1}\!\left(\dfrac{x+y}{1-xy}\right) & \text{if } xy > 1,\; x < 0,\; y < 0 \end{cases}$$
Subtraction
$$\tan^{-1} x - \tan^{-1} y = \tan^{-1}\!\left(\frac{x - y}{1 + xy}\right), \quad xy > -1$$
D
Double Angle Formulae (2 tan⁻¹)
Converting $2\tan^{-1}x$ into $\sin^{-1}$, $\cos^{-1}$, or $\tan^{-1}$
Three equivalent forms
$$2\tan^{-1} x = \sin^{-1}\!\left(\frac{2x}{1+x^2}\right), \quad |x| \leq 1$$
$$2\tan^{-1} x = \cos^{-1}\!\left(\frac{1-x^2}{1+x^2}\right), \quad x \geq 0$$
$$2\tan^{-1} x = \tan^{-1}\!\left(\frac{2x}{1-x^2}\right), \quad |x| < 1$$
✦
Complete Summary — Set II
| Property | Formula | Condition |
|---|---|---|
| Negation (sin) | $\sin^{-1}(-x)=-\sin^{-1}x$ | $x\in[-1,1]$ |
| Negation (cos) | $\cos^{-1}(-x)=\pi-\cos^{-1}x$ | $x\in[-1,1]$ |
| Negation (tan) | $\tan^{-1}(-x)=-\tan^{-1}x$ | $x\in\mathbb{R}$ |
| Negation (cot) | $\cot^{-1}(-x)=\pi-\cot^{-1}x$ | $x\in\mathbb{R}$ |
| Negation (sec) | $\sec^{-1}(-x)=\pi-\sec^{-1}x$ | $|x|\geq1$ |
| Negation (csc) | $\csc^{-1}(-x)=-\csc^{-1}x$ | $|x|\geq1$ |
| Complementary (sin+cos) | $\sin^{-1}x+\cos^{-1}x=\pi/2$ | $x\in[-1,1]$ |
| Complementary (tan+cot) | $\tan^{-1}x+\cot^{-1}x=\pi/2$ | $x\in\mathbb{R}$ |
| Complementary (sec+csc) | $\sec^{-1}x+\csc^{-1}x=\pi/2$ | $|x|\geq1$ |
| tan⁻¹ Addition (xy<1) | $\tan^{-1}x+\tan^{-1}y=\tan^{-1}\!\left(\frac{x+y}{1-xy}\right)$ | $xy<1$ |
| tan⁻¹ Subtraction | $\tan^{-1}x-\tan^{-1}y=\tan^{-1}\!\left(\frac{x-y}{1+xy}\right)$ | $xy>-1$ |
| Double angle (sin⁻¹) | $2\tan^{-1}x=\sin^{-1}\!\left(\frac{2x}{1+x^2}\right)$ | $|x|\leq1$ |
| Double angle (cos⁻¹) | $2\tan^{-1}x=\cos^{-1}\!\left(\frac{1-x^2}{1+x^2}\right)$ | $x\geq0$ |
| Double angle (tan⁻¹) | $2\tan^{-1}x=\tan^{-1}\!\left(\frac{2x}{1-x^2}\right)$ | $|x|<1$ |