(C) Let $t = \tan \theta$. Then $\theta = \tan^{-1} t$. $x = \cos^{-1}\left(\frac{1}{\sqrt{1+\tan^2 \theta}}\right) = \cos^{-1}\left(\frac{1}{\sec \th

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(C) Let $t = \tan \theta$. Then $\theta = \tan^{-1} t$. $x = \cos^{-1}\left(\frac{1}{\sqrt{1+\tan^2 \theta}}\right) = \cos^{-1}\left(\frac{1}{\sec \theta}\right) = \cos^{-1}(\cos \theta) = \theta$. $y = \sin^{-1}\left(\frac{\tan \theta}{\sqrt{1+\tan^2 \theta}}\right) = \sin^{-1}\left(\frac{\tan \theta}{\sec \theta}\right) = \sin^{-1}(\sin \theta) = \theta$. Since $x = \theta$ and $y = \theta$,we have $y = x$. Therefore,$\frac{dy}{dx} = \frac{d}{dx}(x) = 1$.

Class 12 Mathematics Inverse Trigonometric Functions Mix Examples-ITF

Difficult

If $x=\cos ^{-1}\left(\frac{1}{\sqrt{1+t^2}}\right)$ and $y=\sin ^{-1}\left(\frac{t}{\sqrt{1+t^2}}\right)$,then $\frac{dy}{dx}$ is

MHT CET 2022 All MHT CET

A
$0$
B
$\frac{\sin t}{\cos t}$
C
$1$
D
$\sin t \cdot \cos t$

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$\lim _{x \rightarrow 1} \frac{\sqrt{x}-1}{(\cos ^{-1} x)^2} = $

MediumAP EAMCET 2024

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The equation $\cos ^{-1}(1-x)-2 \cos ^{-1} x=\frac{\pi}{2}$ has

DifficultAP EAMCET 2025

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The possible values of $x$, which satisfy the trigonometric equation $\tan ^{-1}\left(\frac{x-1}{x-2}\right)+\tan ^{-1}\left(\frac{x+1}{x+2}\right)=\frac{\pi}{4}$ are

MediumWBJEE 2017

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Match List $I$ with List $II$ and select the correct answer using the code given below the lists:

List $I$ List $II$

$P$. $\left(\frac{1}{y^2}\left(\frac{\cos (\tan ^{-1} y)+y \sin (\tan ^{-1} y)}{\cot (\sin ^{-1} y)+\tan (\sin ^{-1} y)}\right)^2+y^4\right)^{1 / 2}$ takes value $1$. $\frac{1}{2} \sqrt{\frac{5}{3}}$

$Q$. If $\cos x+\cos y+\cos z=0=\sin x+\sin y+\sin z$ then possible value of $\cos \frac{x-y}{2}$ is $2$. $\sqrt{2}$

$R$. If $\cos (\frac{\pi}{4}-x) \cos 2 x+\sin x \sin 2 x \sec x=\cos x \sin 2 x \sec x+\cos (\frac{\pi}{4}+x) \cos 2 x$ then possible value of $\sec x$ is $3$. $\frac{1}{2}$

$S$. If $\cot (\sin ^{-1} \sqrt{1-x^2})=\sin (\tan ^{-1}(x \sqrt{6})), x \neq 0$, then possible value of $x$ is $4$. $1$

Codes: $P \quad Q \quad R \quad S$

List $I$	List $II$
$P$. $\left(\frac{1}{y^2}\left(\frac{\cos (\tan ^{-1} y)+y \sin (\tan ^{-1} y)}{\cot (\sin ^{-1} y)+\tan (\sin ^{-1} y)}\right)^2+y^4\right)^{1 / 2}$ takes value	$1$. $\frac{1}{2} \sqrt{\frac{5}{3}}$
$Q$. If $\cos x+\cos y+\cos z=0=\sin x+\sin y+\sin z$ then possible value of $\cos \frac{x-y}{2}$ is	$2$. $\sqrt{2}$
$R$. If $\cos (\frac{\pi}{4}-x) \cos 2 x+\sin x \sin 2 x \sec x=\cos x \sin 2 x \sec x+\cos (\frac{\pi}{4}+x) \cos 2 x$ then possible value of $\sec x$ is	$3$. $\frac{1}{2}$
$S$. If $\cot (\sin ^{-1} \sqrt{1-x^2})=\sin (\tan ^{-1}(x \sqrt{6})), x \neq 0$, then possible value of $x$ is	$4$. $1$