# Properties of Inverse Trigonometric Functions – II

## Properties of Inverse Trigonometric Functions – II

Property VII:

(i) {{\sin }^{-1}}x+{{\sin }^{-1}}y=\left\{ \begin{align} & {{\sin }^{-1}}\left( x\sqrt{1-{{y}^{2}}}+y\sqrt{1-{{x}^{2}}} \right),\,if\,-1\le x,y\le 1\,and\,{{x}^{2}}+{{y}^{2}}\le 1 \\ &\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, or \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if\,xy<0\,and\,{{x}^{2}}+{{y}^{2}}>1 \\ & \pi -{{\sin }^{-1}}\left( x\sqrt{1-{{y}^{2}}}+y\sqrt{1-{{x}^{2}}} \right),\,if\,0<x,y\le 1\,and\,{{x}^{2}}+{{y}^{2}}>1 \\ & -p-{{\sin }^{-1}}\left( x\sqrt{1-{{y}^{2}}}+y\sqrt{1-{{x}^{2}}} \right),\,if\,-1\le x,y<0\,and\,{{x}^{2}}+{{y}^{2}}>1 \\ \end{align} \right..

(ii) {{\sin }^{-1}}x-{{\sin }^{-1}}y=\left\{ \begin{align} & {{\sin }^{-1}}\,\left( x\sqrt{1-{{y}^{2}}}-y\sqrt{1-{{x}^{2}}} \right),\,if\,-1\le \,x,y\le \,1\,and\,{{x}^{2}}+{{y}^{2}}\le 1 \\ &\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, or \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if\,xy>0\,and\,{{x}^{2}}+{{y}^{2}}>\,1 \\ & \pi -{{\sin }^{-1}}\left( x\sqrt{1-{{y}^{2}}}-y\sqrt{1-{{x}^{2}}} \right),\,if\,0<x\le 1,-1\le y\le 0\,and\,{{x}^{2}}+{{y}^{2}}>1 \\ & -\pi -{{\sin }^{-1}}\left( x\sqrt{1-{{y}^{2}}} \right)-y\sqrt{1-{{x}^{2}}},\,if\,-1\le x<0,0<y\le1\,and\,{{x}^{2}}+{{y}^{2}}>1 \\ \end{align} \right..

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..

Example 1: Find the value of $${{\cot }^{-1}}\left( \frac{3}{4} \right)+{{\sin }^{-1}}\left( \frac{5}{13} \right)$$.

Solution: $${{\cot }^{-1}}\left( \frac{3}{4} \right)+{{\sin }^{-1}}\left( \frac{5}{13} \right)$$.

= $${{\sin }^{-1}}\left( \frac{4}{5} \right)+{{\sin }^{-1}}\left( \frac{5}{13} \right)$$.

= $${{\sin }^{-1}}\left( \frac{4}{5}\sqrt{1-{{\left( \frac{5}{13} \right)}^{2}}}+\frac{5}{13}\sqrt{1-{{\left( \frac{4}{5} \right)}^{2}}} \right)$$.

= $${{\sin }^{-1}}\left( \frac{4}{5}\times \frac{12}{13}+\frac{5}{13}\times \frac{3}{5} \right)$$.

$$={{\sin }^{-1}}\left( \frac{63}{65} \right)$$.

Example 2: Solve $${{\sin }^{-1}}x+{{\sin }^{-1}}2x=\frac{\pi }{3}$$.

Solution: Given that

$${{\sin }^{-1}}x+{{\sin }^{-1}}2x=\frac{\pi }{3}$$.

$${{\sin }^{-1}}2x={{\sin }^{-1}}\left( \frac{\sqrt{3}}{2} \right)-{{\sin }^{-1}}x$$.

= $${{\sin }^{-1}}\left( \frac{\sqrt{3}}{2}\sqrt{1-{{x}^{2}}}-x\sqrt{1-\frac{3}{4}} \right)$$.

$$2x=\frac{\sqrt{3}}{2}\sqrt{1-{{x}^{2}}}-\frac{x}{2}$$.

$$\left( \frac{5{{x}^{2}}}{2} \right)=\frac{3}{4}(1-{{x}^{2}})$$.

28 x² = 3

$$x=\sqrt{\frac{3}{28}}$$.