Advertisements
Advertisements
Question
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`
Solution
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
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`
If (tan−1x)2 + (cot−1x)2 = 5π2/8, then find x.
`sin^-1(sin (5pi)/6)`
Evaluate the following:
`sec^-1{sec (-(7pi)/3)}`
Write the following in the simplest form:
`tan^-1(x/(a+sqrt(a^2-x^2))),-a<x<a`
Evaluate the following:
`sin(sin^-1 7/25)`
Prove the following result
`cos(sin^-1 3/5+cot^-1 3/2)=6/(5sqrt13)`
Evaluate:
`cot{sec^-1(-13/5)}`
Evaluate:
`sin(tan^-1x+tan^-1 1/x)` for x > 0
`sin^-1x=pi/6+cos^-1x`
`4sin^-1x=pi-cos^-1x`
Prove the following result:
`sin^-1 12/13+cos^-1 4/5+tan^-1 63/16=pi`
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`
Evaluate the following:
`sin(2tan^-1 2/3)+cos(tan^-1sqrt3)`
`tan^-1 2/3=1/2tan^-1 12/5`
Solve the following equation for x:
`tan^-1((x-2)/(x-1))+tan^-1((x+2)/(x+1))=pi/4`
Write the value ofWrite the value of \[2 \sin^{- 1} \frac{1}{2} + \cos^{- 1} \left( - \frac{1}{2} \right)\]
If \[\sin^{- 1} \left( \frac{1}{3} \right) + \cos^{- 1} x = \frac{\pi}{2},\] then find x.
If x < 0, y < 0 such that xy = 1, then write the value of tan−1 x + tan−1 y.
If \[\cos\left( \sin^{- 1} \frac{2}{5} + \cos^{- 1} x \right) = 0\], find the value of x.
Find the value of \[\cos^{- 1} \left( \cos\frac{13\pi}{6} \right)\]
If \[\cos^{- 1} \frac{x}{a} + \cos^{- 1} \frac{y}{b} = \alpha, then\frac{x^2}{a^2} - \frac{2xy}{ab}\cos \alpha + \frac{y^2}{b^2} = \]
The number of solutions of the equation \[\tan^{- 1} 2x + \tan^{- 1} 3x = \frac{\pi}{4}\] is
The number of real solutions of the equation \[\sqrt{1 + \cos 2x} = \sqrt{2} \sin^{- 1} (\sin x), - \pi \leq x \leq \pi\]
\[\cot\left( \frac{\pi}{4} - 2 \cot^{- 1} 3 \right) =\]
If x > 1, then \[2 \tan^{- 1} x + \sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\] is equal to
