\displaystyle \textbf{1. }(i)\ \sin^{-1}(\sin \theta)=\theta,\ \text{for all }\theta\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]
\displaystyle (ii)\ \cos^{-1}(\cos \theta)=\theta,\ \text{for all }\theta\in\left[0,\pi\right]
\displaystyle (iii)\ \tan^{-1}(\tan \theta)=\theta,\ \text{for all }\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)
\displaystyle (iv)\ \mathrm{cosec}^{-1}(\mathrm{cosec}\ \theta)=\theta,\ \text{for all }\theta\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right],\ \theta\ne0
\displaystyle (v)\ \sec^{-1}(\sec \theta)=\theta,\ \text{for all }\theta\in\left[0,\pi\right],\ \theta\ne\frac{\pi}{2}
\displaystyle (vi)\ \cot^{-1}(\cot \theta)=\theta,\ \text{for all }\theta\in\left(0,\pi\right)

\displaystyle \textbf{2. }(i)\ \sin(\sin^{-1}x)=x,\ \text{for all }x\in\left[-1,1\right]
\displaystyle (ii)\ \cos(\cos^{-1}x)=x,\ \text{for all }x\in\left[-1,1\right]
\displaystyle (iii)\ \tan(\tan^{-1}x)=x,\ \text{for all }x\in R
\displaystyle (iv)\ \mathrm{cosec}(\mathrm{cosec}^{-1}x)=x,\ \text{for all }x\in(-\infty,-1]\cup[1,\infty)
\displaystyle (v)\ \sec(\sec^{-1}x)=x,\ \text{for all }x\in(-\infty,-1]\cup[1,\infty)
\displaystyle (vi)\ \cot(\cot^{-1}x)=x,\ \text{for all }x\in R
\displaystyle \text{Note: }\text{It should be noted that }\sin^{-1}(\sin \theta)\ne\theta,\ \text{if }\theta\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right].
\displaystyle \text{In fact, we have}
\displaystyle \sin^{-1}(\sin \theta)=\begin{cases}-\pi-\theta,&\text{if }\theta\in\left[-\frac{3\pi}{2},-\frac{\pi}{2}\right]\\\theta,&\text{if }\theta\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\\\pi-\theta,&\text{if }\theta\in\left[\frac{\pi}{2},\frac{3\pi}{2}\right]\\-2\pi+\theta,&\text{if }\theta\in\left[\frac{3\pi}{2},\frac{5\pi}{2}\right]\end{cases}\ \text{and so on.}
\displaystyle \text{Similarly, we have}
\displaystyle \cos^{-1}(\cos \theta)=\begin{cases}-\theta,&\text{if }\theta\in[-\pi,0]\\\theta,&\text{if }\theta\in[0,\pi]\\2\pi-\theta,&\text{if }\theta\in[\pi,2\pi]\\-2\pi+\theta,&\text{if }\theta\in[2\pi,3\pi]\end{cases}\ \text{and so on.}
\displaystyle \tan^{-1}(\tan \theta)=\begin{cases}\pi+\theta,&\text{if }\theta\in\left(-\frac{3\pi}{2},-\frac{\pi}{2}\right)\\\theta,&\text{if }\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\\\theta-\pi,&\text{if }\theta\in\left(\frac{\pi}{2},\frac{3\pi}{2}\right)\\\theta-2\pi,&\text{if }\theta\in\left(\frac{3\pi}{2},\frac{5\pi}{2}\right)\end{cases}\ \text{and so on.}

\displaystyle \textbf{3. }(i)\ \sin^{-1}(-x)=-\sin^{-1}x,\ \text{for all }x\in[-1,1]
\displaystyle (ii)\ \cos^{-1}(-x)=\pi-\cos^{-1}x,\ \text{for all }x\in[-1,1]
\displaystyle (iii)\ \tan^{-1}(-x)=-\tan^{-1}x,\ \text{for all }x\in R
\displaystyle (iv)\ \mathrm{cosec}^{-1}(-x)=-\mathrm{cosec}^{-1}x,\ \text{for all }x\in(-\infty,-1]\cup[1,\infty)
\displaystyle (v)\ \sec^{-1}(-x)=\pi-\sec^{-1}x,\ \text{for all }x\in(-\infty,-1]\cup[1,\infty)
\displaystyle (vi)\ \cot^{-1}(-x)=\pi-\cot^{-1}x,\ \text{for all }x\in R

\displaystyle \textbf{4. }(i)\ \sin^{-1}\left(\frac{1}{x}\right)=\mathrm{cosec}^{-1}x,\ \text{for all }x\in(-\infty,-1]\cup[1,\infty)
\displaystyle (ii)\ \cos^{-1}\left(\frac{1}{x}\right)=\sec^{-1}x,\ \text{for all }x\in(-\infty,-1]\cup[1,\infty)
\displaystyle (iii)\ \tan^{-1}\left(\frac{1}{x}\right)=\begin{cases}\cot^{-1}x,&\text{for }x>0\\-\pi+\cot^{-1}x,&\text{for }x<0\end{cases}

\displaystyle \textbf{5. }(i)\ \sin^{-1}x+\cos^{-1}x=\frac{\pi}{2},\ \text{for all }x\in[-1,1]
\displaystyle (ii)\ \tan^{-1}x+\cot^{-1}x=\frac{\pi}{2},\ \text{for all }x\in R
\displaystyle (iii)\ \sec^{-1}x+\mathrm{cosec}^{-1}x=\frac{\pi}{2},\ \text{for all }x\in(-\infty,-1]\cup[1,\infty)

\displaystyle \textbf{6. }(i)\ \tan^{-1}x+\tan^{-1}y=\begin{cases}\tan^{-1}\left(\frac{x+y}{1-xy}\right),&\text{if }xy<1\\\pi+\tan^{-1}\left(\frac{x+y}{1-xy}\right),&\text{if }x>0,y>0\text{ and }xy>1\\-\pi+\tan^{-1}\left(\frac{x+y}{1-xy}\right),&\text{if }x<0,y<0\text{ and }xy>1\end{cases}
\displaystyle (ii)\ \tan^{-1}x-\tan^{-1}y=\begin{cases}\tan^{-1}\left(\frac{x-y}{1+xy}\right),&\text{if }xy>-1\\\pi+\tan^{-1}\left(\frac{x-y}{1+xy}\right),&\text{if }x>0,y<0\text{ and }xy<-1\\-\pi+\tan^{-1}\left(\frac{x-y}{1+xy}\right),&\text{if }x<0,y>0\text{ and }xy<-1\end{cases}
\displaystyle \text{Note: }\text{If }x_1,x_2,x_3,\ldots,x_n\in R,\text{ then}
\displaystyle \tan^{-1}x_1+\tan^{-1}x_2+\cdots+\tan^{-1}x_n=\tan^{-1}\left(\frac{S_1-S_3+S_5-S_7+\cdots}{1-S_2+S_4-S_6+\cdots}\right),
\displaystyle \text{where }S_k\text{ denotes the sum of the products of }x_1,x_2,\ldots,x_n\text{ taken }k\text{ at a time.}

\displaystyle \textbf{7. }(i)\ \sin^{-1}x+\sin^{-1}y \\ =\begin{cases}\sin^{-1}\left(x\sqrt{1-y^2}+y\sqrt{1-x^2}\right),&\text{if }-1\le x,y\le1\text{ and }x^2+y^2\le1\\\pi-\sin^{-1}\left(x\sqrt{1-y^2}+y\sqrt{1-x^2}\right),&\text{if }0<x,y\le1\text{ and }x^2+y^2>1\\-\pi-\sin^{-1}\left(x\sqrt{1-y^2}+y\sqrt{1-x^2}\right),&\text{if }-1\le x,y<0\text{ and }x^2+y^2>1\end{cases}
\displaystyle (ii)\ \sin^{-1}x-\sin^{-1}y \\ =\begin{cases}\sin^{-1}\left(x\sqrt{1-y^2}-y\sqrt{1-x^2}\right),&\text{if }-1\le x,y\le1\text{ and }x^2+y^2\le1\\\pi-\sin^{-1}\left(x\sqrt{1-y^2}-y\sqrt{1-x^2}\right),&\text{if }0<x\le1,-1\le y<0\text{ and }x^2+y^2>1\\-\pi-\sin^{-1}\left(x\sqrt{1-y^2}-y\sqrt{1-x^2}\right),&\text{if }-1\le x<0,0<y\le1\text{ and }x^2+y^2>1\end{cases}

\displaystyle \textbf{8. }(i)\ \cos^{-1}x+\cos^{-1}y \\ =\begin{cases}\cos^{-1}\left(xy-\sqrt{1-x^2}\sqrt{1-y^2}\right),&\text{if }-1\le x,y\le1\text{ and }x+y\ge0\\2\pi-\cos^{-1}\left(xy-\sqrt{1-x^2}\sqrt{1-y^2}\right),&\text{if }-1\le x,y\le1\text{ and }x+y\le0\end{cases}
\displaystyle (ii)\ \cos^{-1}x-\cos^{-1}y \\ =\begin{cases}\cos^{-1}\left(xy+\sqrt{1-x^2}\sqrt{1-y^2}\right),&\text{if }-1\le x,y\le1\text{ and }x\le y\\-\cos^{-1}\left(xy+\sqrt{1-x^2}\sqrt{1-y^2}\right),&\text{if }-1\le y\le0,0<x\le1\text{ and }x\ge y\end{cases}

\displaystyle \textbf{9. }(i)\ 2\sin^{-1}x=\begin{cases}\sin^{-1}\left(2x\sqrt{1-x^2}\right),&\text{if }-\frac{1}{\sqrt{2}}\le x\le\frac{1}{\sqrt{2}}\\\pi-\sin^{-1}\left(2x\sqrt{1-x^2}\right),&\text{if }\frac{1}{\sqrt{2}}\le x\le1\\-\pi-\sin^{-1}\left(2x\sqrt{1-x^2}\right),&\text{if }-1\le x\le-\frac{1}{\sqrt{2}}\end{cases}
\displaystyle (ii)\ 3\sin^{-1}x=\begin{cases}\sin^{-1}\left(3x-4x^3\right),&\text{if }-\frac{1}{2}\le x\le\frac{1}{2}\\\pi-\sin^{-1}\left(3x-4x^3\right),&\text{if }\frac{1}{2}<x\le1\\-\pi-\sin^{-1}\left(3x-4x^3\right),&\text{if }-1\le x<-\frac{1}{2}\end{cases}

\displaystyle \textbf{10. }(i)\ 2\cos^{-1}x=\begin{cases}\cos^{-1}\left(2x^2-1\right),&\text{if }0\le x\le1\\2\pi-\cos^{-1}\left(2x^2-1\right),&\text{if }-1\le x\le0\end{cases}
\displaystyle (ii)\ 3\cos^{-1}x=\begin{cases}\cos^{-1}\left(4x^3-3x\right),&\text{if }\frac{1}{2}\le x\le1\\2\pi-\cos^{-1}\left(4x^3-3x\right),&\text{if }-\frac{1}{2}\le x\le\frac{1}{2}\\2\pi+\cos^{-1}\left(4x^3-3x\right),&\text{if }-1\le x\le-\frac{1}{2}\end{cases}

\displaystyle \textbf{11. }(i)\ 2\tan^{-1}x=\begin{cases}\tan^{-1}\left(\frac{2x}{1-x^2}\right),&\text{if }-1<x<1\\\pi+\tan^{-1}\left(\frac{2x}{1-x^2}\right),&\text{if }x>1\\-\pi+\tan^{-1}\left(\frac{2x}{1-x^2}\right),&\text{if }x<-1\end{cases}
\displaystyle (ii)\ 3\tan^{-1}x=\begin{cases}\tan^{-1}\left(\frac{3x-x^3}{1-3x^2}\right),&\text{if }-\frac{1}{\sqrt{3}}<x<\frac{1}{\sqrt{3}}\\\pi+\tan^{-1}\left(\frac{3x-x^3}{1-3x^2}\right),&\text{if }x>\frac{1}{\sqrt{3}}\\-\pi+\tan^{-1}\left(\frac{3x-x^3}{1-3x^2}\right),&\text{if }x<-\frac{1}{\sqrt{3}}\end{cases}

\displaystyle \textbf{12. }(i)\ 2\tan^{-1}x= \left\{\begin{array}{ll}\sin^{-1}\left(\frac{2x}{1+x^2}\right),&\text{if }-1\le x\le1\\\pi-\sin^{-1}\left(\frac{2x}{1+x^2}\right),&\text{if }x>1\\-\pi-\sin^{-1}\left(\frac{2x}{1+x^2}\right),&\text{if }x<-1\end{array}\right.
\displaystyle (ii)\ 2\tan^{-1}x= \left\{\begin{array}{ll}\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right),&\text{if }0\le x<\infty\\-\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right),&\text{if }-\infty<x\le0\end{array}\right.

\displaystyle \textbf{13. }(i)\ \sin^{-1}x=\cos^{-1}\sqrt{1-x^2}=\tan^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right)
\displaystyle =\cot^{-1}\left(\frac{\sqrt{1-x^2}}{x}\right)=\sec^{-1}\left(\frac{1}{\sqrt{1-x^2}}\right)=\mathrm{cosec}^{-1}\left(\frac{1}{x}\right)
\displaystyle (ii)\ \cos^{-1}x=\sin^{-1}\sqrt{1-x^2}=\tan^{-1}\left(\frac{\sqrt{1-x^2}}{x}\right)
\displaystyle =\cot^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right)=\sec^{-1}\left(\frac{1}{x}\right)=\mathrm{cosec}^{-1}\left(\frac{1}{\sqrt{1-x^2}}\right)
\displaystyle (iii)\ \tan^{-1}x=\sin^{-1}\left(\frac{x}{\sqrt{1+x^2}}\right)=\cos^{-1}\left(\frac{1}{\sqrt{1+x^2}}\right)
\displaystyle =\cot^{-1}\left(\frac{1}{x}\right)=\sec^{-1}\sqrt{1+x^2}=\mathrm{cosec}^{-1}\left(\frac{\sqrt{1+x^2}}{x}\right)

\displaystyle \textbf{14. }\text{If }x_1,x_2,\ldots,x_n\in R,\text{ then}
\displaystyle \tan^{-1}x_1+\tan^{-1}x_2+\cdots+\tan^{-1}x_n=\tan^{-1}\left(\frac{S_1-S_3+S_5-S_7+\cdots}{1-S_2+S_4-S_6+\cdots}\right),\ \text{where}
\displaystyle S_k=\text{Sum of the products of }x_1,x_2,\ldots,x_n\ \text{taken }k\ \text{at a time.}

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