# Properties of Inverse Trigonometric Functions with Example

Here, you will learn all the properties of inverse trigonometric functions class 12 with examples.

Let’s begin –

## Properties of Inverse Trigonometric functions

Property – 1

(i)  y = $$sin^{-1}(sinx)$$, x $$\in$$ R, y $$\in$$ (-$$\pi\over 2$$, $$\pi\over 2$$) periodic with period $$2\pi$$
and it is an odd function.

(ii)  y = $$cos^{-1}(cosx)$$, x $$\in$$ R, y $$\in$$ [0, $$\pi$$], periodic with period $$2\pi$$ and it is an even function.

(iii)  y = $$tan^{-1}(tanx)$$, x $$\in$$ R – { (2n-1)$$\pi\over 2$$, n $$\in$$ I }, y $$\in$$ (-$$\pi\over 2$$, $$\pi\over 2$$) periodic with period $$\pi$$ and it is an odd function.

(iv)  y = $$cot^{-1}(cotx)$$, x $$\in$$ R – { n$$\pi$$, n $$\in$$ I }, y $$\in$$ (0, $$\pi$$) periodic with period $$\pi$$ and neither even or odd function.

(v)  y = $$cosec^{-1}(cosecx)$$, x $$\in$$ R – { n$$\pi$$, n $$\in$$ I }, y $$\in$$ [-$$\pi\over 2$$, 0] $$\cup$$ (0, $$\pi\over 2$$] periodic with period $$2\pi$$ and it is an odd function.

(vi)  y = $$sec^{-1}(secx)$$, x $$\in$$ R – { (2n-1)$$\pi\over 2$$, n $$\in$$ I }, y $$\in$$ [0, $$\pi\over 2$$] $$\cup$$ ($$\pi\over 2$$, $$\pi$$], y is periodic with period $$2\pi$$ and it is an even function.

Example : Evaluate $$sin^{-1}(sin10)$$

Solution : We know that $$sin^{-1}(sinx)$$ = x, if $$-\pi\over 2$$ $$\le$$ x $$\le$$ $$\pi\over 2$$

Here, x = 10 radians which does not lie between -$$\pi\over 2$$ and $$\pi\over 2$$

But, $$3\pi$$ – x i.e. $$3\pi$$ – 10 lie between -$$\pi\over 2$$ and $$\pi\over 2$$

Also, sin($$3\pi$$ – 10) = sin 10

$$\therefore$$   $$sin^{-1}(sin10)$$ = $$sin^{-1}(sin(3\pi – 10)$$ = ($$3\pi$$ – 10)

Property – 2

(i)  $$sin^{-1}x$$ + $$cos^{-1}x$$ = $$\pi\over 2$$

(ii)  $$tan^{-1}x$$ + $$cot^{-1}x$$ = $$\pi\over 2$$

(iii)  $$cosec^{-1}x$$ + $$sec^{-1}x$$ = $$\pi\over 2$$

Property – 3

(i)   $$sin^{-1}(-x)$$ = -$$sin^{-1}x$$

(ii)  $$cosec^{-1}(-x)$$ = -$$cosec^{-1}x$$

(iii)  $$tan^{-1}(-x)$$ = -$$tan^{-1}x$$

(iv)  $$cot^{-1}(-x)$$ = $$\pi$$ – $$cot^{-1}x$$

(v)  $$cos^{-1}(-x)$$ = $$\pi$$ – $$cos^{-1}x$$

(vi) $$sec^{-1}(-x)$$ = $$\pi$$ – $$sec^{-1}x$$

Property – 4

(i)  $$cosec^{-1}x$$ = $$sin^{-1}{1\over x}$$

(ii)  $$sec^{-1}x$$ = $$cos^{-1}{1\over x}$$

(iii)  $$cot^{-1}x$$ = $$\begin{cases} tan^{-1}{1\over x}, & \text{if}\ x > 0 \\ \pi + tan^{-1}{1\over x}, & \text{if}\ x < 0 \end{cases}$$

Example : Find the value of x if $$cos^{-1}(-x)$$ + $$tan^{-1}(-x)$$ – 2$$sin^{-1}x$$ + $$sec^{-1}({-1\over x})$$ = $$\pi\over 4$$ for |x| $$\le$$ 1.

Solution : $$\pi$$ – $$cos^{-1}(x)$$ – $$tan^{-1}(x)$$ – 2$$sin^{-1}x$$ + $$cos^{-1}(-x)$$ = $$\pi\over 4$$

$$\pi$$ – $$cos^{-1}(x)$$ – $$tan^{-1}(x)$$ – 2$$sin^{-1}x$$ + $$\pi$$ – $$cos^{-1}(-x)$$ = $$\pi\over 4$$

2$$\pi$$ – 2($$sin^{-1}x$$ + $$cos^{-1}x$$) – $$\pi\over 4$$ = $$tan^{-1}(x)$$

2$$\pi$$ – $$\pi$$ – $$\pi\over 4$$ = $$tan^{-1}(x)$$ $$\implies$$ $$tan^{-1}(x)$$ = $$3\pi\over 4$$   Hence no solution.

Property – 5

(i)

(a)   $$tan^{-1}x$$ + $$tan^{-1}y$$ = $$\begin{cases} tan^{-1}{{x+y}\over {1-xy}}, & \text{where}\ x > 0, y > 0 & xy < 1 \\ \pi + tan^{-1}{{x+y}\over {1-xy}}, & \text{where}\ x > 0, y > 0 & xy > 1 \\ {\pi\over 2} , & \text{where}\ x > 0, y > 0 & xy = 1 \end{cases}$$

(b)  $$tan^{-1}x$$ – $$tan^{-1}y$$ = $$tan^{-1}{{x-y}\over {1+xy}}$$

(c)  $$tan^{-1}x$$ + $$tan^{-1}y$$ + $$tan^{-1}z$$ = $$tan^{-1}[{{x+y+z-xyz}\over {1-xy-yz-zx}}]$$

(ii)

(a)  $$sin^{-1}x$$ + $$sin^{-1}y$$ = $$\begin{cases} sin^{-1}[{x\sqrt{1-y^2} + y{\sqrt{1-x^2}}}], & \text{where}\ x > 0, y > 0 & (x^2 + y^2) \le 1 \\ \pi – sin^{-1}[{x\sqrt{1-y^2} + y{\sqrt{1-x^2}}}], & \text{where}\ x > 0, y > 0 & (x^2 + y^2) > 1 \end{cases}$$

(b)  $$sin^{-1}x$$ – $$sin^{-1}y$$ = $$sin^{-1}[{x\sqrt{1-y^2} – y{\sqrt{1-x^2}}}]$$, where x > 0, y > 0

(iii)

(a)  $$cos^{-1}x$$ + $$cos^{-1}y$$ = $$cos^{-1}[xy – {\sqrt{1-y^2}{\sqrt{1-x^2}}}]$$, where x > 0, y > 0

(b)  $$cos^{-1}x$$ – $$cos^{-1}y$$ = $$\begin{cases} cos^{-1}[xy + {\sqrt{1-y^2}{\sqrt{1-x^2}}}]; x < y, \ x, y > 0 \\ – cos^{-1}[xy + {\sqrt{1-y^2}{\sqrt{1-x^2}}}], x > y, \ x, y > 0 \end{cases}$$

Example : Prove that : $$tan^{-1}{1\over 7}$$ + $$tan^{-1}{1\over 13}$$ = $$tan^{-1}{2\over 9}$$

Solution : L.H.S = $$tan^{-1}{1\over 7}$$ + $$tan^{-1}{1\over 13}$$

= $$tan^{-1}[{{{1\over 7}+{1\over 13}}\over {1 – {1\over 7}\times{1\over 13}}}]$$       { $$\because$$ $$tan^{-1}x$$ + $$tan^{-1}y$$ = $$tan^{-1}{{x+y}\over {1-xy}}$$; if xy < 1 }

= $$tan^{-1}({20\over 90})$$ = $$tan^{-1}({2\over 9})$$ = R.H.S.