Domains and Range of Inverse Trigonometric:

PYQ-2023-ITF-Q1

Important Identitites:

PYQ-2023-ITF-Q6 , PYQ-2023-Definite_Integration-Q11

• $ \sin \left(\sin ^{-1}(x)\right)=x; \quad -1 \leq x \leq 1$

• $ \cos ^{-1}(\cos x)=x ; \quad 0 \leq x \leq \pi$

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

• $ \cot ^{-1}(\cot \mathrm{x})=\mathrm{x} ; \quad 0<\mathrm{x}<\pi$

• $ \sec ^{-1}(\sec x)=x ; \quad 0 \leq x \leq \pi, x \neq \frac{\pi}{2}$

• $ \operatorname{cosec}^{-1}(\operatorname{cosec} x)=x ; \quad x \neq 0,-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$

• $ \sin ^{-1}(-x)=-\sin ^{-1} x, \quad-1 \leq x \leq 1$

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

• $ \cos ^{-1}(-x)=\pi-\cos ^{-1} x, \quad-1 \leq x \leq 1$

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

• $ \sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2},-1 \leq x \leq 1$

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

• $ \operatorname{cosec}^{-1} x+\sec ^{-1} x=\frac{\pi}{2},|x| \geq 1$

• $ \sin ^{-1}(1 / x)=\operatorname{cosec}^{-1}(x)$, if $x \geq 1$ or $x \leq-1$

• $ \cos ^{-1}(1 / x)=\sec ^{-1}(x)$, if $x \geq 1$ or $x \leq-1$

• $ \tan ^{-1}(1 / x)=\cot ^{-1}(x), x>0$

Identities of Addition and Substraction:

PYQ-2023-ITF-Q3 , PYQ-2023-ITF-Q4 , PYQ-2023-ITF-Q8

• $ \sin ^{-1} x+\sin ^{-1} y=\sin ^{-1}\left[x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}\right], x \geq 0, y \geq 0 \hspace{1mm}$&${\hspace{1mm} \left(x^{2}+y^{2}\right) \leq 1}$

$\quad \quad \quad \quad \sin ^{-1} x+\sin ^{-1} y=\pi-\sin ^{-1}\left[x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}\right], x \geq 0, y \geq 0 \hspace{1mm} \hspace{1mm} x^{2}+y^{2}>1$

• $ \cos ^{-1} x+\cos ^{-1} y=\cos ^{-1}\left[x y-\sqrt{1-x^{2}} \sqrt{1-y^{2}}\right], x \geq 0, y \geq 0$

• $ \tan ^{-1} x+\tan ^{-1} y=\tan ^{-1} \frac{x+y}{1-x y}, x>0, y>0 \hspace{1mm} $ & $ {\hspace{1mm} x y<1}$

$\quad \quad \quad \quad \tan ^{-1} x+\tan ^{-1} y = \pi+\tan ^{-1} \frac{x+y}{1-x y}, x>0, y>0 \hspace{1mm} $ & $ \hspace{1mm} x y>1=\frac{\pi}{2}, x>0, y>0\hspace{1mm}\hspace{1mm} x y=1 $

• $ \sin ^{-1} x-\sin ^{-1} y=\sin ^{-1}\left[x \sqrt{1-y^{2}}-y \sqrt{1-x^{2}}\right], x \geq 0, y \geq 0$

• $ \cos ^{-1} x-\cos ^{-1} y=\cos ^{-1}\left[x y+\sqrt{1-x^{2}} \sqrt{1-y^{2}}\right], x \geq 0, y \geq 0, x \leq y $

• $ \tan ^{-1} x-\tan ^{-1} y=\tan ^{-1} \frac{x-y}{1+x y}, x \geq 0, y \geq 0$

• $ \tan ^{-1} x+\tan ^{-1} y+\tan ^{-1} z=\tan ^{-1}\left[\frac{x+y+z-x y z}{1-x y-y z-z x}\right]$ if, $x>0, y>0, z>0 \hspace{1mm}$ & $\hspace{1mm}(x y+y z+z x)<1$

Double Angle Formulae:

PYQ-2023-ITF-Q2 , PYQ-2023-ITF-Q7 , PYQ-2023-ITF-Q8

• $\sin^{-1}(2x\sqrt{1-x^2}) = \begin{cases} 2\sin^{-1}x & \text{if } |x| \leq \frac{1}{\sqrt{2}} \\ \pi - 2\sin^{-1}x & \text{if } x > \frac{1}{\sqrt{2}} \\ -(\pi + 2\sin^{-1}x) & \text{if } x < -\frac{1}{\sqrt{2}} \end{cases}$

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

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

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

• $\cos^{-1}\frac{1-x^2}{1+x^2} = \begin{cases} 2\tan^{-1}x & \text{if } x \geq 0 \\ -2\tan^{-1}x & \text{if } x < 0 \end{cases}$

Triple Angle Formulae:

• $\quad 3 \sin ^{-1} x=\sin ^{-1}\left(3 x-4 x^3\right)$

• $\quad 3 \cos ^{-1} x=\cos ^{-1}\left(4 x^3-3 x\right)$

• $\quad 3 \tan ^{-1} x=\tan ^{-1}\left(\frac{3 x-x^3}{1-3 x^2}\right)$