Advertisements
Advertisements
प्रश्न
Prove that
`tan^(-1) [(sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x))]=pi/4-1/2 cos^(-1)x,-1/sqrt2<=x<=1`
Advertisements
उत्तर
To prove
`tan^(-1) [(sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x))]=pi/4-1/2 cos^(-1)x,-1/sqrt2<=x<=1`
Taking LHS, we get:
`tan^(-1) [(sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x))]`
let `x=cos 2theta`
`tan^(-1) [(sqrt(1+x)-sqrt(1-x))/(sqrt(1+cos2theta)+sqrt(1-cos2theta))]=tan^(-1) [(sqrt(1+cos2theta)-sqrt(1-cos2theta))/(sqrt(1+cos2theta)+sqrt(1-cos2theta))]`
`=tan^(-1)[(costheta-sintheta)/(costheta+sintheta)]`
`=tan^(-1)[(1-tantheta)/(1+tantheta)]`
`=tan^(-1) tan(pi/4-theta)`
`=(pi/4-theta)`
`=π/4−θ`
`=π/4−1/2cos^(−1) x`
`=RHS `
Hence proved.
APPEARS IN
संबंधित प्रश्न
If a line makes angles 90° and 60° respectively with the positive directions of x and y axes, find the angle which it makes with the positive direction of z-axis.
Evaluate the following:
`cos^-1{cos ((4pi)/3)}`
Evaluate the following:
`tan^-1(tan (7pi)/6)`
Evaluate the following:
`\text(cosec)^-1(\text{cosec} pi/4)`
Evaluate the following:
`sin(sin^-1 7/25)`
Evaluate:
`cos(tan^-1 3/4)`
Evaluate:
`sin(tan^-1x+tan^-1 1/x)` for x > 0
Prove the following result:
`tan^-1 1/4+tan^-1 2/9=sin^-1 1/sqrt5`
If `cos^-1 x/2+cos^-1 y/3=alpha,` then prove that `9x^2-12xy cosa+4y^2=36sin^2a.`
Prove that: `cos^-1 4/5+cos^-1 12/13=cos^-1 33/65`
Evaluate the following:
`sin(1/2cos^-1 4/5)`
`tan^-1 1/4+tan^-1 2/9=1/2cos^-1 3/2=1/2sin^-1(4/5)`
`2sin^-1 3/5-tan^-1 17/31=pi/4`
Prove that:
`tan^-1 (2ab)/(a^2-b^2)+tan^-1 (2xy)/(x^2-y^2)=tan^-1 (2alphabeta)/(alpha^2-beta^2),` where `alpha=ax-by and beta=ay+bx.`
For any a, b, x, y > 0, prove that:
`2/3tan^-1((3ab^2-a^3)/(b^3-3a^2b))+2/3tan^-1((3xy^2-x^3)/(y^3-3x^2y))=tan^-1 (2alphabeta)/(alpha^2-beta^2)`
`where alpha =-ax+by, beta=bx+ay`
If x > 1, then write the value of sin−1 `((2x)/(1+x^2))` in terms of tan−1 x.
If x < 0, then write the value of cos−1 `((1-x^2)/(1+x^2))` in terms of tan−1 x.
Write the value of tan−1x + tan−1 `(1/x)`for x > 0.
Write the value of
\[\cos^{- 1} \left( \frac{1}{2} \right) + 2 \sin^{- 1} \left( \frac{1}{2} \right)\].
Write the value of cos−1 \[\left( \tan\frac{3\pi}{4} \right)\]
Write the value of tan−1\[\left\{ \tan\left( \frac{15\pi}{4} \right) \right\}\]
Write the value of cos−1 \[\left( \cos\frac{5\pi}{4} \right)\]
If \[\tan^{- 1} (\sqrt{3}) + \cot^{- 1} x = \frac{\pi}{2},\] find x.
Write the principal value of \[\sin^{- 1} \left\{ \cos\left( \sin^{- 1} \frac{1}{2} \right) \right\}\]
Find the value of \[\cos^{- 1} \left( \cos\frac{13\pi}{6} \right)\]
If sin−1 x − cos−1 x = `pi/6` , then x =
If x < 0, y < 0 such that xy = 1, then tan−1 x + tan−1 y equals
The value of \[\cos^{- 1} \left( \cos\frac{5\pi}{3} \right) + \sin^{- 1} \left( \sin\frac{5\pi}{3} \right)\] is
The value of sin \[\left( \frac{1}{4} \sin^{- 1} \frac{\sqrt{63}}{8} \right)\] is
