Advertisements
Advertisements
Question
Show that \[\sin^{- 1} (2x\sqrt{1 - x^2}) = 2 \sin^{- 1} x\]
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 value of the following: `tan(1/2)[sin^(-1)((2x)/(1+x^2))+cos^(-1)((1-y^2)/(1+y^2))],|x| <1,y>0 and xy <1`
Solve the following for x :
`tan^(-1)((x-2)/(x-3))+tan^(-1)((x+2)/(x+3))=pi/4,|x|<1`
Solve the following for x:
`sin^(-1)(1-x)-2sin^-1 x=pi/2`
If tan-1x+tan-1y=π/4,xy<1, then write the value of x+y+xy.
`sin^-1(sin pi/6)`
`sin^-1(sin4)`
Evaluate the following:
`cos^-1(cos4)`
Evaluate the following:
`cos^-1(cos12)`
Evaluate the following:
`tan^-1(tan (9pi)/4)`
Evaluate the following:
`tan^-1(tan1)`
Evaluate the following:
`tan^-1(tan4)`
Evaluate the following:
`sec^-1(sec (13pi)/4)`
Evaluate the following:
`cot^-1{cot (-(8pi)/3)}`
Write the following in the simplest form:
`tan^-1sqrt((a-x)/(a+x)),-a<x<a`
Evaluate the following:
`sin(cos^-1 5/13)`
Evaluate the following:
`sin(sec^-1 17/8)`
Evaluate the following:
`cot(cos^-1 3/5)`
Evaluate:
`cosec{cot^-1(-12/5)}`
`4sin^-1x=pi-cos^-1x`
`5tan^-1x+3cot^-1x=2x`
Prove the following result:
`tan^-1 1/7+tan^-1 1/13=tan^-1 2/9`
`2sin^-1 3/5-tan^-1 17/31=pi/4`
`2tan^-1(1/2)+tan^-1(1/7)=tan^-1(31/17)`
Prove that
`sin{tan^-1 (1-x^2)/(2x)+cos^-1 (1-x^2)/(2x)}=1`
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
If `sin^-1x+sin^-1y+sin^-1z=(3pi)/2,` then write the value of x + y + z.
If x > 1, then write the value of sin−1 `((2x)/(1+x^2))` in terms of tan−1 x.
What is the value of cos−1 `(cos (2x)/3)+sin^-1(sin (2x)/3)?`
Write the value of cos\[\left( 2 \sin^{- 1} \frac{1}{3} \right)\]
Evaluate sin \[\left( \tan^{- 1} \frac{3}{4} \right)\]
Write the value of tan−1\[\left\{ \tan\left( \frac{15\pi}{4} \right) \right\}\]
Write the value of \[\sin^{- 1} \left( \frac{1}{3} \right) - \cos^{- 1} \left( - \frac{1}{3} \right)\]
The value of tan \[\left\{ \cos^{- 1} \frac{1}{5\sqrt{2}} - \sin^{- 1} \frac{4}{\sqrt{17}} \right\}\] is
\[\text{ If }\cos^{- 1} \frac{x}{3} + \cos^{- 1} \frac{y}{2} = \frac{\theta}{2}, \text{ then }4 x^2 - 12xy \cos\frac{\theta}{2} + 9 y^2 =\]
It \[\tan^{- 1} \frac{x + 1}{x - 1} + \tan^{- 1} \frac{x - 1}{x} = \tan^{- 1}\] (−7), then the value of x is
In a ∆ ABC, if C is a right angle, then
\[\tan^{- 1} \left( \frac{a}{b + c} \right) + \tan^{- 1} \left( \frac{b}{c + a} \right) =\]
The value of \[\tan\left( \cos^{- 1} \frac{3}{5} + \tan^{- 1} \frac{1}{4} \right)\]
