Inverse trigonometric ratios of Multiple Angles

What are Multiple angles in terms of inverse trigonometric function?

The multiple angle formula is the representation of the angle which is multiple of a given angle. We can calculate the values of multiple angles by expressing each trigonometric function in its expanded form.

Multiple angles in terms of arcsin

1. Double of Inverse Trigonometric Function Formulas

$2 \sin ^{-1} \mathrm{x}=\left\{\begin{array}{cc}\sin ^{-1}\left(2 x \sqrt{1-x^2}\right), & \frac{1}{\sqrt{2}} \leq x \leq \frac{1}{\sqrt{2}} \\ \pi-\sin ^{-1}\left(2 x \sqrt{1-x^2}\right), & x>\frac{1}{\sqrt{2}} \\ -\pi-\sin ^{-1}\left(2 x \sqrt{1-x^2}\right), & x<-\frac{1}{\sqrt{2}}\end{array}\right.$
2. Triple of Inverse Trigonometric Function Formulas

$3 \sin ^{-1} \mathrm{x}=\left\{\begin{array}{cc}\sin ^{-1}\left(3 x-4 x^3\right), & -\frac{1}{2} \leq x \leq \frac{1}{2} \\ \pi-\sin ^{-1}\left(3 x-4 x^3\right), & x>\frac{1}{2} \\ -\pi-\sin ^{-1}\left(3 x-4 x^3\right) & x:-\frac{1}{2}\end{array}\right.$

Multiple angles in terms of arccos

1. Double of Inverse Trigonometric Function Formulas

$2 \cos ^{-1} \mathrm{x}=\left\{\begin{array}{cc}\cos ^{-1}\left(2 x^2-1\right), & \text { if } 0 \leq x \leq 1 \\ 2 \pi-\cos ^{-1}\left(2 x^2-1\right), & \text { if }-1 \leq x \leq 0\end{array}\right.$
2. Triple of Inverse Trigonometric Function Formulas

$3 \cos ^{-1} \mathrm{x}=\left\{\begin{array}{cc}\cos ^{-1}\left(4 x^3-3 x\right), & \text { if } \frac{1}{2} \leq x \leq 1 \\ 2 \pi-\cos ^{-1}\left(4 x^3-3 x\right), & \text { if }-\frac{1}{2} \leq x \leq \frac{1}{2} \\ 2 \pi+\cos ^{-1}\left(4 x^3-3 x\right), & \text { if }-1 \leq x \leq-\frac{1}{1}\end{array}\right.$

Multiple angles in terms of arctan and arcsin

The multiple-angle formula of arctan in terms of arcsin is given by

$2 \tan ^{-1} \mathrm{x}=\left\{\begin{array}{cc}\sin ^{-1}\left(\frac{2 x}{1+x^2}\right), & \text { if }-1 \leq x \leq 1 \\ \pi-\sin ^{-1}\left(\frac{2 x}{1+x^2}\right), & \text { if } x>1 \\ -\pi-\sin ^{-1}\left(\frac{2 x}{1+x^2}\right), & \text { if } x<-1\end{array}\right.$

Multiple angles in terms of arctan and arccos

The multiple angle formula of arctan in terms of arccos is given by

$2 \tan ^{-1} \mathrm{x}=\left\{\begin{array}{cc}\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right), & \text { if } 0 \leq x<\infty \\ -\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right), & \text { if }-\infty<x \leq 0\end{array}\right.$

Recommended Video Based on Inverse Trigonometric of Multiple Angles :

Solved Examples Based on Multiple Angles in Terms of Inverse Trigonometric Functions

Example 1: $S=\left\{x \in R ; 0<x<1\right.$ and $\left.2 \tan ^{-1}\left(\frac{1-x}{1+x}\right)=\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right\}$ If n(S) denotes the number of elements in S then : [JEE MAINS 2023]

Solution: Put $x=\tan \theta \quad \theta \in\left(0, \frac{\pi}{4}\right)$

$\begin{aligned} & 2 \tan ^{-1}\left(\frac{1-\tan \theta}{1+\tan \theta}\right)=\cos ^{-1}\left(\frac{1-\tan ^2 \theta}{1+\tan ^2 \theta}\right) \\ & 2 \tan ^{-1}\left[\tan \left(\frac{\pi}{4}-\theta\right)\right]=\cos ^{-1}[\cos (2 \theta)] \\ & \Rightarrow 2\left(\frac{\pi}{4}-\theta\right)=2 \theta \Rightarrow \theta=\frac{\pi}{8} \\ & \Rightarrow x=\tan \frac{\pi}{8}=\sqrt{2}-1 \simeq 0.414\end{aligned}$

Hence, the answer is $n(S)=2$ and only one element in S is less then $\frac{1}{2}$

Example 2: If the sum of all the solutions of $\tan ^{-1}\left(\frac{2 x}{1-x^2}\right)+\cot ^{-1}\left(\frac{1-x^2}{2 x}\right)=\frac{\pi}{3},-1<x<1, x \neq 0{ }_{\text {is }} \alpha-\frac{4}{\sqrt{3}}$, then $\alpha$ is equal to
[JEE MAINS 2023]

Solution$x \in(-1,1) \quad \tan ^{-1}\left(\frac{2 \mathrm{x}}{1-\mathrm{x}^2}\right)=2 \tan ^{-1} \mathrm{x}$
$x \in(0,1) \quad \cot ^{-1}\left(\frac{1-x^2}{2 x}\right)=\tan ^{-1}\left(\frac{2 x}{1-x^2}\right)=2 \tan ^{-1} x$
$x \in(-1,0) \quad \cot ^{-1}\left(\frac{1-x^2}{2 x}\right)=\mathrm{IT}+\tan ^{-1}\left(\frac{2 x}{1-\mathrm{x}^2}\right)=\mathrm{IT}+2 \tan ^{-1} \mathrm{x}$$x \in(0,1) \quad 2 \tan ^{-1} x+2 \tan ^{-1} x=\pi / 3$

$\tan ^{-1} x=\frac{\pi}{12}$

$\mathrm{x}=2-\sqrt{3}$$\begin{gathered}x \in(-1,0) \quad 2 \tan ^{-1} x+I T+2 \tan ^{-1} x=\pi / 3 \\ 4 \tan ^{-1} x=\frac{-2 \pi}{3} \\ \tan ^{-1} \mathrm{x}=-\pi / 6 \\ \mathrm{x}=-1 / \sqrt{3} \\ (2-\sqrt{3})+\left(-\frac{1}{\sqrt{3}}\right)=a-\frac{4}{\sqrt{3}} \\ 2-4 / \sqrt{3}=\alpha-4 / \sqrt{3} \\ \alpha=2\end{gathered}$
Hence, the answer is 2.

Example 3: $\operatorname{cosec}\left[2 \cot ^{-1}(5)+\cos ^{-1}\left(\frac{4}{5}\right)\right]$ is equal to: [JEE MAINS 2021]

Solution
$
\begin{aligned}
& \operatorname{cosec}\left[2 \tan ^{-1}\left(\frac{1}{5}\right)+\tan ^{-1}\left(\frac{3}{4}\right)\right] \\
& \operatorname{cosec}\left[\tan ^{-1}\left(\frac{5}{12}\right)+\tan ^{-1}\left(\frac{3}{4}\right)\right] \\
& \operatorname{cosec}\left[\tan ^{-1}\left(\frac{\frac{5}{12}+\frac{3}{4}}{1-\frac{5}{12} \times \frac{3}{4}}\right)\right] \\
& =\operatorname{cosec}\left[\tan ^{-1}\left(\frac{56}{33}\right)\right]=\frac{65}{56}
\end{aligned}
$
$
\text { Hence, the answer is } \frac{65}{56}
$

Example 4: The sum of possible values of x for $\tan ^{-1}(x+1)+\cot ^{-1}\left(\frac{1}{x-1}\right)=\tan ^{-1}\left(\frac{8}{31}\right)$ is : [JEE MAINS 2021]

Solution
$
\tan ^{-1}(x+1)+\cot ^{-1}\left(\frac{1}{x-1}\right)=\tan ^{-1} \frac{8}{31}
$
Taking tangent both sides :-

$
\begin{aligned}
& \frac{(x+1)+(x-1)}{1-\left(x^2-1\right)}=\frac{8}{31} \\
& \Rightarrow \frac{2 x}{2-x^2}=\frac{8}{31} \\
& \Rightarrow 4 x^2+31 x-8=0 \\
& \Rightarrow x=-8, \frac{1}{4}
\end{aligned}
$

$\begin{aligned} & \text { But, if } x=\frac{1}{4} \\ & \tan ^{-1}(x+1) \in\left(0, \frac{\pi}{2}\right) \\ & \& \cot ^{-1}\left(\frac{1}{x-1}\right) \in\left(\frac{\pi}{2}, \pi\right)\end{aligned}$
$
\Rightarrow L H S>\frac{\pi}{2} \quad \& \quad R H S<\frac{\pi}{2}
$

(Not possible)
Hence, $x=-8$

Hence, the answer is $-\frac{32}{4}$

Example 5: If $2 y=\left(\cot ^{-1}\left(\frac{\sqrt{3} \cos x+\sin x}{\cos x-\sqrt{3} \sin x}\right)\right)^2, x \in\left(0, \frac{\pi}{2}\right)$ $\square$ then $\overline{d x}$ is equal to [JEE MAINS 2019]

Solution: Important Results of Inverse Trigonometric Functions $\tan ^{-1} x+\cot ^{-1} x=\frac{\pi}{2}$ when $x \in R$

$2 y=\left(\cot ^{-1}\left(\frac{\sqrt{3} \cos x+\sin x}{\cos x-\sqrt{3} \sin x}\right)\right)^2 \quad x \in\left(0, \frac{\pi}{2}\right)$

$2 y=\left(\cot ^{-1}\left(\frac{\frac{\sqrt{3} \cos x}{\cos x}+\frac{\sin x}{\cos x}}{\frac{\cos x}{\cos x}-\frac{\sqrt{3} \sin x}{\cos x}}\right)\right)^2$

$=\left(\cot ^{-1}\left(\frac{\sqrt{3}+\tan x}{1-\sqrt{3} \tan x}\right)\right)^2$

$=\left(\cot ^{-1}\left(\tan \left(\frac{\pi}{3}+x\right)\right)\right)^2$

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

$=\left(\frac{\pi}{2}-\left(\frac{\pi}{3}+x\right)\right)^2$

$=\left(\frac{\pi}{2}-\frac{\pi}{3}-x\right)^2=\left(\frac{\pi}{6}-x\right)^2$

$2 y=\left(x-\frac{\pi}{6}\right)^2$

$\frac{d y}{d x}=\frac{1}{2} \times 2\left(x-\frac{\pi}{6}\right)$

$=x-\frac{\pi}{6}$

Hence, the answer is $x-\frac{\pi}{6}$