Property VIII:
(i) \({{\cos }^{-1}}x+{{\cos }^{-1}}y=\left\{ \begin{align}& {{\cos }^{-1}}\left( xy-\sqrt{1-{{x}^{2}}}\sqrt{1-{{y}^{2}}} \right),if\,-1\le x,\,y\le 1\,and\,x+y\ge 0 \\& 2\pi -{{\cos }^{-1}}\left( xy-\sqrt{1-{{x}^{2}}}\sqrt{1-{{y}^{2}}} \right),if\,-1\le x,y\le 1\,and\,x+y\le 0 \\\end{align} \right.\).
(ii) \({{\cos }^{-1}}x-{{\cos }^{-1}}y=\left\{ \begin{align}& {{\cos }^{-1}}\left( xy+\sqrt{1-{{x}^{2}}}\sqrt{1-{{y}^{2}}} \right),if\,-1\le x,y\le 1\,and\,x\le y \\& -{{\cos }^{-1}}\left( xy+\sqrt{1-{{x}^{2}}}\sqrt{1-{{y}^{2}}} \right),if\,-1\le y\le 0,\,0<x\le 1\,and\,x\ge y \\\end{align} \right.\).
Property IX:
(i) \(2{{\sin }^{-1}}x=\left\{ \begin{align}& {{\sin }^{-1}}\left( 2x\sqrt{1-{{x}^{2}}} \right),if\,-\frac{1}{\sqrt{2}}\le x\le \frac{1}{\sqrt{2}} \\& \pi -{{\sin }^{-1}}92x\sqrt{1-{{x}^{2}}},if\,\frac{1}{\sqrt{2}}\le x\le 1 \\& -\pi -\sin \left( 2x\sqrt{1-{{x}^{2}}} \right),if\,-1\le x\le -\frac{1}{\sqrt{2}} \\\end{align} \right.\).
(ii) \(3{{\sin }^{-1}}x=\left\{ \begin{align}& {{\sin }^{-1}}\,\left( 3x-4{{x}^{3}} \right),if\,\frac{1}{2}\le x\le \frac{1}{2} \\& \pi -{{\sin }^{-1}}\left( 3-4{{x}^{3}} \right),if\,\frac{1}{2}<x\le 1 \\& -\pi -{{\sin }^{-1}}\left( 3x-4{{x}^{3}} \right),if\,-1\le x<-\frac{1}{2} \\\end{align} \right.\).
Property X:
(i) \(2{{\cos }^{-1}}x=\left\{ \begin{align}& {{\cos }^{-1}}\left( 2{{x}^{2}}-1 \right),if\,0\le x\le 1 \\& 2\pi -{{\cos }^{-1}}\left( 2{{x}^{2}}-1 \right),if\,-1\le x\le 0 \\\end{align} \right.\).
(ii) \(3{{\cos }^{-1}}x=\left\{ \begin{align}& 2\pi -{{\cos }^{-1}}\left( 4{{x}^{3}}-3x \right),if\,\frac{1}{2}\le x\le 1 \\& 2\pi -{{\cos }^{-1}}\left( 4{{x}^{3}}-3x \right),if\,-\frac{1}{2}\le \frac{1}{2} \\& 2\pi +{{\cos }^{-1}}\left( 4{{x}^{3}}-3x \right),if\,-1\le x\le -\frac{1}{2} \\\end{align} \right.\).
Property XI:
(i) \(2{{\tan }^{-1}}x=\left\{ \begin{align}& {{\tan }^{-1}}\left( \frac{2x}{1-{{x}^{2}}} \right),if\,-1<x<1 \\& \pi +{{\tan }^{-1}}\left( \frac{2x}{1-{{x}^{2}}} \right),if\,x>1 \\& -\pi +{{\tan }^{-1}}\left( \frac{2x}{1-{{x}^{2}}} \right),if\,x<-1 \\\end{align} \right.\).
(ii) \(3{{\tan }^{-1}}x=\left\{ \begin{align}& {{\tan }^{-1}}\left( \frac{3x-{{x}^{3}}}{1-3{{x}^{2}}} \right),if\,-\frac{1}{\sqrt{3}}<x<\frac{1}{\sqrt{3}} \\& \pi +{{\tan }^{-1}}\left( \frac{3x-{{x}^{3}}}{1-3{{x}^{2}}} \right),if\,x>\frac{1}{\sqrt{3}} \\& -\pi +{{\tan }^{-1}}\left( \frac{3x-{{x}^{3}}}{1-3{{x}^{2}}} \right),if\,x<-\frac{1}{\sqrt{3}} \\\end{align} \right.\).
Property XII:
(i) \(2{{\tan }^{-1}}x=\left\{ \begin{align}& {{\sin }^{-1}}\left( \frac{2x}{1+{{x}^{2}}} \right),if\,-1\le x\le 1 \\& \pi -{{\sin }^{-1}}\left( \frac{2x}{1+{{x}^{2}}} \right),if\,x>1 \\& -\pi -{{\sin }^{-1}}\left( \frac{2x}{1+{{x}^{2}}} \right),if\,x<-1 \\\end{align} \right.\).
(ii) \(2{{\tan }^{-1}}x=\left\{ \begin{align}& {{\cos }^{-1}}\left( \frac{1-{{x}^{2}}}{1+{{x}^{2}}} \right),if\,0\le x\le \infty \\& -{{\cos }^{-1}}\left( \frac{1-{{x}^{2}}}{1+{{x}^{2}}} \right),if\,-\infty <x\le 0 \\\end{align} \right.\).
Property XIII:
(i) sin–¹ x = cos–¹ √(1 – x²) = tan–¹ [x / √(1 – x²)]
= cot–¹ √(1 – x²) / x = sec–¹ [1 / √(1 – x²)]
= cosec–¹ (1/x)
(ii) cos–¹ x = sin–¹ √(1 – x²) = tan–¹ [√(1 – x²) / x]
= cot–¹ [x / √(1 – x²)] = sec–¹ 1/x
= cosec–¹ [1/ √(1 – x²)]
(iii) tan–¹ x = sin–¹ [x / √(1 + x²)] = cos–¹ [x / √(1 + x²)]
= cot–¹ (1/x) = sec–¹ √(1 + x²)
= cosec–¹ [√(1 + x²) / x]
Property XIV: If x₁, x₂, … xn ϵ R, then
tan–¹ x₁ + tan–¹ x₂ + … + tan–¹ xn = tan–¹ [(s₁ – s₃ + s₅ – s₇ + …) / (1 – s₂ + s₄ – s₆ + …)]
Where Sk = sum of the products of x₁, x₂, … xn taken k at a time.