If sin ^{-1}(1-x)-2 sin ^{-1} x=frac{pi}{2},t | vedclass

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

Question

(B) Given the equation: $\sin ^{-1}(1-x)-2 \sin ^{-1} x=\frac{\pi}{2}$.Let $\sin ^{-1} x = 	heta$,then $x = \sin 	heta$. Since $x \in [-1, 1]$,$	he

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(B) Given the equation: $\sin ^{-1}(1-x)-2 \sin ^{-1} x=\frac{\pi}{2}$.Let $\sin ^{-1} x = 	heta$,then $x = \sin 	heta$. Since $x \in [-1, 1]$,$	heta \in [-\frac{\pi}{2}, \frac{\pi}{2}]$.The equation becomes $\sin ^{-1}(1-x) = \frac{\pi}{2} + 2	heta$.Taking $\sin$ on both sides: $1-x = \sin(\frac{\pi}{2} + 2	heta) = \cos(2	heta)$.Using the identity $\cos(2	heta) = 1 - 2\sin^2 	heta$,we get $1-x = 1 - 2x^2$.This simplifies to $2x^2 - x = 0$,which means $x(2x-1) = 0$.Thus,$x = 0$ or $x = \frac{1}{2}$.Checking $x = 0$: $\sin^{-1}(1) - 2\sin^{-1}(0) = \frac{\pi}{2} - 0 = \frac{\pi}{2}$. This is a solution.Checking $x = \frac{1}{2}$: $\sin^{-1}(1-\frac{1}{2}) - 2\sin^{-1}(\frac{1}{2}) = \sin^{-1}(\frac{1}{2}) - 2(\frac{\pi}{6}) = \frac{\pi}{6} - \frac{\pi}{3} = -\frac{\pi}{6} 
eq \frac{\pi}{2}$.Therefore,the only solution is $x = 0$.

If $\sin ^{-1}(1-x)-2 \sin ^{-1} x=\frac{\pi}{2}$,then $x=$ . . . . . . .

Similar Questions

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

If ${\sin ^{ - 1}}a + {\sin ^{ - 1}}b + {\sin ^{ - 1}}c = \pi ,$ then the value of $a\sqrt {1 - {a^2}} + b\sqrt {1 - {b^2}} + c\sqrt {1 - {c^2}}$ is

The derivative of ${\sin ^{ - 1}}\left( {\frac{{2x}}{{1 + {x^2}}}} \right)$ with respect to ${\cos ^{ - 1}}\left( {\frac{{1 - {x^2}}}{{1 + {x^2}}}} \right)$ is

$\sum\limits_{\lambda = 1}^{10} {{{\sin }^{ - 1}}\left( {\sin \left( {\lambda \pi - \frac{\pi }{6}} \right)} \right)} $ is equal to-

The complete solution set of the inequality $(\sec^{-1}x - 4)(\sec^{-1}x - 1)(\sec^{-1}x - 2) \ge 0$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module