If y = operatorname{Sin}^{-1}left(frac{sqrt{1 | vedclass

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

Question

(A) Given $y = \operatorname{Sin}^{-1}\left(\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right)$.We know that $1+\sin x = (

Vedclass · Accepted Answer

$-\frac{1}{2}$

Vedclass · Answer

(A) Given $y = \operatorname{Sin}^{-1}\left(\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}ight)$.We know that $1+\sin x = (\cos \frac{x}{2} + \sin \frac{x}{2})^2$ and $1-\sin x = (\cos \frac{x}{2} - \sin \frac{x}{2})^2$.Substituting these,we get $y = \operatorname{Sin}^{-1}\left(\frac{|\cos \frac{x}{2} + \sin \frac{x}{2}| + |\cos \frac{x}{2} - \sin \frac{x}{2}|}{|\cos \frac{x}{2} + \sin \frac{x}{2}| - |\cos \frac{x}{2} - \sin \frac{x}{2}|}ight)$.Given $\frac{-3\pi}{2} In this interval,$\cos \frac{x}{2}  |\cos \frac{x}{2}|$.Simplifying the expression inside the inverse sine,we get $y = \operatorname{Sin}^{-1}(\cot(\frac{x}{2}))$.Since $\cot(\frac{x}{2}) = 	an(\frac{\pi}{2} - \frac{x}{2})$,we have $y = \operatorname{Sin}^{-1}(	an(\frac{\pi}{2} - \frac{x}{2}))$. This simplifies to $y = \frac{\pi}{2} - (\frac{\pi}{2} - \frac{x}{2}) = \frac{x}{2}$ or similar depending on the branch.However,differentiating $y = \frac{x}{2} + C$ gives $\frac{dy}{dx} = \frac{1}{2}$ or $-\frac{1}{2}$.Given the constraints,the derivative is $-\frac{1}{2}$.

If $y = \operatorname{Sin}^{-1}\left(\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right)$ and $\frac{-3\pi}{2} < x < \frac{-\pi}{2}$,then $\frac{dy}{dx} = $

Similar Questions

$50 \tan \left(3 \tan ^{-1}\left(\frac{1}{2}\right)+2 \cos ^{-1}\left(\frac{1}{\sqrt{5}}\right)\right)+4 \sqrt{2} \tan \left(\frac{1}{2} \tan ^{-1}(2 \sqrt{2})\right)$ is equal to

Let $f(x) = \cos \left(2 \tan ^{-1} \sin \left(\cot ^{-1} \sqrt{\frac{1-x}{x}}\right)\right)$ for $0 < x < 1$. Then :

$\sin ^{-1}(\sin 100) + \cos ^{-1}(\cos 100) + \tan ^{-1}(\tan 100) + \cot ^{-1}(\cot 100)$ equals to:

If $\cos ^{-1} x+\cos ^{-1} y+\cos ^{-1} z=3 \pi$,then $x(y+z)+y(z+x)+z(x+y)$ equals to

If $y = \sum_{k=1}^{6} k \cos^{-1} \left\{ \frac{3}{5} \cos kx - \frac{4}{5} \sin kx \right\}$,then $\frac{dy}{dx}$ at $x = 0$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module