Advertisements
Advertisements
Question
Show that \[\sin^{- 1} (2x\sqrt{1 - x^2}) = 2 \sin^{- 1} x\]
Advertisements
Solution
We have
\[LHS = \sin^{- 1} \left( 2x\sqrt{1 - x^2} \right)\]
\[\text{Putting }x = \sin a, \text{we get}\]
\[ = \sin^{- 1} \left( 2 \sin a\sqrt{1 - \sin^2 a} \right) \]
\[ = \sin^{- 1} \left( 2\sin a \cos a \right)\]
\[ = \sin^{- 1} \left( \sin 2a \right)\]
\[ = 2a\]
\[ = 2 \sin^{- 1} x \left( \because x = \sin a \right)\]
\[\]
APPEARS IN
RELATED QUESTIONS
Find the domain of `f(x) =2cos^-1 2x+sin^-1x.`
Find the principal values of the following:
`cos^-1(-sqrt3/2)`
`sin^-1(sin pi/6)`
`sin^-1(sin (13pi)/7)`
`sin^-1(sin3)`
`sin^-1(sin12)`
Evaluate the following:
`cos^-1(cos12)`
Evaluate the following:
`tan^-1(tan12)`
Evaluate the following:
`sec^-1(sec (25pi)/6)`
Evaluate the following:
`\text(cosec)^-1(\text{cosec} pi/4)`
Evaluate the following:
`cot^-1(cot (9pi)/4)`
Write the following in the simplest form:
`tan^-1{x+sqrt(1+x^2)},x in R `
Write the following in the simplest form:
`tan^-1{(sqrt(1+x^2)+1)/x},x !=0`
Write the following in the simplest form:
`tan^-1sqrt((a-x)/(a+x)),-a<x<a`
Write the following in the simplest form:
`sin{2tan^-1sqrt((1-x)/(1+x))}`
Evaluate the following:
`tan(cos^-1 8/17)`
Evaluate:
`tan{cos^-1(-7/25)}`
Evaluate:
`cot(sin^-1 3/4+sec^-1 4/3)`
Solve the following equation for x:
tan−1(x + 2) + tan−1(x − 2) = tan−1 `(8/79)`, x > 0
`sin^-1 5/13+cos^-1 3/5=tan^-1 63/16`
If `cos^-1 x/2+cos^-1 y/3=alpha,` then prove that `9x^2-12xy cosa+4y^2=36sin^2a.`
`tan^-1 1/7+2tan^-1 1/3=pi/4`
Find the value of the following:
`tan^-1{2cos(2sin^-1 1/2)}`
Find the value of the following:
`cos(sec^-1x+\text(cosec)^-1x),` | x | ≥ 1
Solve the following equation for x:
`tan^-1((2x)/(1-x^2))+cot^-1((1-x^2)/(2x))=(2pi)/3,x>0`
Write the value of sin (cot−1 x).
Write the value of sin−1
\[\left( \sin( -{600}°) \right)\].
Write the value of cos2 \[\left( \frac{1}{2} \cos^{- 1} \frac{3}{5} \right)\]
Write the value of tan−1\[\left\{ \tan\left( \frac{15\pi}{4} \right) \right\}\]
Write the value ofWrite the value of \[2 \sin^{- 1} \frac{1}{2} + \cos^{- 1} \left( - \frac{1}{2} \right)\]
Evaluate: \[\sin^{- 1} \left( \sin\frac{3\pi}{5} \right)\]
Write the value of \[\cos^{- 1} \left( \cos\frac{14\pi}{3} \right)\]
Write the principal value of \[\sin^{- 1} \left\{ \cos\left( \sin^{- 1} \frac{1}{2} \right) \right\}\]
The set of values of `\text(cosec)^-1(sqrt3/2)`
2 tan−1 {cosec (tan−1 x) − tan (cot−1 x)} is equal to
The positive integral solution of the equation
\[\tan^{- 1} x + \cos^{- 1} \frac{y}{\sqrt{1 + y^2}} = \sin^{- 1} \frac{3}{\sqrt{10}}\text{ is }\]
sin \[\left\{ 2 \cos^{- 1} \left( \frac{- 3}{5} \right) \right\}\] is equal to
\[\cot\left( \frac{\pi}{4} - 2 \cot^{- 1} 3 \right) =\]
Write the value of \[\cos^{- 1} \left( - \frac{1}{2} \right) + 2 \sin^{- 1} \left( \frac{1}{2} \right)\] .
The value of sin `["cos"^-1 (7/25)]` is ____________.
