Given 0 leq x leq frac{1}{2},then the value o | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let $\sin^{-1} x = 	heta$,which implies $x = \sin 	heta$. Since $0 \leq x \leq \frac{1}{2}$,we have $0 \leq 	heta \leq \frac{\pi}{6}$.Then $\sq

Vedclass · Accepted Answer

$1$

Vedclass · Answer

(A) Let $\sin^{-1} x = 	heta$,which implies $x = \sin 	heta$. Since $0 \leq x \leq \frac{1}{2}$,we have $0 \leq 	heta \leq \frac{\pi}{6}$.Then $\sqrt{1-x^2} = \cos 	heta$.The expression becomes $	an \left[ \sin^{-1} \left( \frac{\sin 	heta}{\sqrt{2}} + \frac{\cos 	heta}{\sqrt{2}} ight) - 	heta ight]$.Using the identity $\sin 	heta + \cos 	heta = \sqrt{2} \sin \left( 	heta + \frac{\pi}{4} ight)$,we get:$\frac{\sin 	heta + \cos 	heta}{\sqrt{2}} = \sin \left( 	heta + \frac{\pi}{4} ight)$.Substituting this into the expression:$= 	an \left[ \sin^{-1} \left( \sin \left( 	heta + \frac{\pi}{4} ight) ight) - 	heta ight]$.Since $0 \leq 	heta \leq \frac{\pi}{6}$,then $\frac{\pi}{4} \leq 	heta + \frac{\pi}{4} \leq \frac{\pi}{6} + \frac{\pi}{4} = \frac{5\pi}{12}$.Since this range is within $[-\frac{\pi}{2}, \frac{\pi}{2}]$,$\sin^{-1}(\sin(	heta + \frac{\pi}{4})) = 	heta + \frac{\pi}{4}$.Thus,the expression simplifies to $	an \left( 	heta + \frac{\pi}{4} - 	heta ight) = 	an \frac{\pi}{4} = 1$.

Given $0 \leq x \leq \frac{1}{2}$,then the value of $\tan \left(\sin ^{-1}\left(\frac{x}{\sqrt{2}}+\frac{\sqrt{1-x^2}}{\sqrt{2}}\right)-\sin ^{-1} x\right)$ is

Similar Questions

Where is the largest newsprint paper industry located in India?

Two batteries,one of $emf$ $18 \, V$ and internal resistance $2 \, \Omega$ and the other of $emf$ $12 \, V$ and internal resistance $1 \, \Omega$,are connected in parallel as shown in the figure. The voltmeter $V$ will record a reading of ............. $V$.

Sodium and copper have work functions $2.3 \ eV$ and $4.5 \ eV$ respectively. The ratio of the threshold wavelength of sodium to that of copper is nearest to:

Which of the following reagents are required to prepare $3$-ethylpentan-$3$-ol?

The volume-temperature graphs of a given mass of an ideal gas at constant pressures are shown below. What is the correct order of pressures?

Vedclass Test Series

Exam Paper Generator

Online Exam Module