Advertisements
Advertisements
प्रश्न
Write the following in the simplest form:
`sin^-1{(sqrt(1+x)+sqrt(1-x))/2},0<x<1`
Advertisements
उत्तर
Let x = cos θ
Now,
`sin^-1{(sqrt(1+x)+sqrt(1-x))/2} = sin^-1 {(sqrt(1+costheta)+sqrt(1-costheta))/2}`
`=sin^-1{(sqrt(2cos^2 theta/2)+sqrt(2sin^2 theta/2))/2}`
`=sin^-1{(cos theta/2+sin theta/2)/sqrt2}`
`=sin^-1{1/sqrt2sin theta/2+1/sqrt2cos theta/2}`
`=sin^-1{sin(theta/2+pi/4)}`
`=theta/2+pi/4`
`=(cos^-1x)/2+pi/4`
`therefore sin^-1{(sqrt(1+x)+sqrt(1-x))/2}=(cos^-1x)/2+pi/4`
APPEARS IN
संबंधित प्रश्न
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`
If `cos^-1( x/a) +cos^-1 (y/b)=alpha` , prove that `x^2/a^2-2(xy)/(ab) cos alpha +y^2/b^2=sin^2alpha`
Prove that :
`2 tan^-1 (sqrt((a-b)/(a+b))tan(x/2))=cos^-1 ((a cos x+b)/(a+b cosx))`
Find the domain of `f(x)=cos^-1x+cosx.`
`sin^-1(sin (13pi)/7)`
Evaluate the following:
`cos^-1{cos(-pi/4)}`
Evaluate the following:
`tan^-1(tan4)`
Evaluate the following:
`cosec^-1{cosec (-(9pi)/4)}`
Evaluate the following:
`cot^-1{cot (-(8pi)/3)}`
Evaluate the following:
`sin(tan^-1 24/7)`
Evaluate the following:
`sec(sin^-1 12/13)`
Solve: `cos(sin^-1x)=1/6`
Evaluate:
`cot{sec^-1(-13/5)}`
Evaluate:
`cosec{cot^-1(-12/5)}`
Evaluate:
`cot(sin^-1 3/4+sec^-1 4/3)`
Evaluate:
`sin(tan^-1x+tan^-1 1/x)` for x > 0
Evaluate:
`cos(sec^-1x+\text(cosec)^-1x)`,|x|≥1
Solve the following equation for x:
tan−1(x + 1) + tan−1(x − 1) = tan−1`8/31`
Solve the following equation for x:
`tan^-1((1-x)/(1+x))-1/2 tan^-1x` = 0, where x > 0
Solve the following equation for x:
`tan^-1 x/2+tan^-1 x/3=pi/4, 0<x<sqrt6`
`sin^-1 4/5+2tan^-1 1/3=pi/2`
`2sin^-1 3/5-tan^-1 17/31=pi/4`
`2tan^-1 1/5+tan^-1 1/8=tan^-1 4/7`
`2tan^-1 3/4-tan^-1 17/31=pi/4`
Prove that
`sin{tan^-1 (1-x^2)/(2x)+cos^-1 (1-x^2)/(2x)}=1`
Solve the following equation for x:
`cos^-1((x^2-1)/(x^2+1))+1/2tan^-1((2x)/(1-x^2))=(2x)/3`
Solve the following equation for x:
`tan^-1((x-2)/(x-1))+tan^-1((x+2)/(x+1))=pi/4`
Prove that `2tan^-1(sqrt((a-b)/(a+b))tan theta/2)=cos^-1((a costheta+b)/(a+b costheta))`
Write the value of tan−1x + tan−1 `(1/x)`for x > 0.
If −1 < x < 0, then write the value of `sin^-1((2x)/(1+x^2))+cos^-1((1-x^2)/(1+x^2))`
Write the value of sin (cot−1 x).
Evaluate sin
\[\left( \frac{1}{2} \cos^{- 1} \frac{4}{5} \right)\]
Write the value of \[\tan^{- 1} \left( \frac{1}{x} \right)\] for x < 0 in terms of `cot^-1x`
If \[\cos\left( \sin^{- 1} \frac{2}{5} + \cos^{- 1} x \right) = 0\], find the value of x.
Find the value of \[\cos^{- 1} \left( \cos\frac{13\pi}{6} \right)\]
2 tan−1 {cosec (tan−1 x) − tan (cot−1 x)} is equal to
Let f (x) = `e^(cos^-1){sin(x+pi/3}.`
Then, f (8π/9) =
The value of \[\sin\left( 2\left( \tan^{- 1} 0 . 75 \right) \right)\] is equal to
