Advertisements
Advertisements
Question
If `sin^-1 (2a)/(1+a^2)-cos^-1 (1-b^2)/(1+b^2)=tan^-1 (2x)/(1-x^2)`, then prove that `x=(a-b)/(1+ab)`
Solution
Let: a = tan m
b = tan n
x = tan y
Now,
`sin^-1 (2a)/(1+a^2)-cos^-1 (1-b^2)/(1+b^2)=tan^-1 (2x)/(1-x^2)`
`=>sin^-1 (2tanm)/(1+tan^2m)-cos^-1 (1-tan^2n)/(1+tan^2n)=tan^-1 (2tany)/(1-tan^2y)`
`=>sin^-1(sin2m)-cos^-1(cos2n)=tan^-1(tan2y)` `[becausesin2x=(2tanx)/(1+tan^2x)andcos2x=(1-tan^2x)/(1+tan^2x)]`
`=>2m-2n=2y`
`=>m-n=y`
`=>tan^-1a-tan^-1b=tan^-1x` `[becausea=tanm,b=tannandx=tany]`
`=>tan^-1 (a-b)/(1+ab)=tan^-1x` `[becausetan^-1x-tan^-1y=tan^-1 (x-y)/(1+xy)]``=>(a-b)/(1+ab)=x`
`therefore(a-b)/(1+ab)=x`
APPEARS IN
RELATED QUESTIONS
Solve the equation for x:sin−1x+sin−1(1−x)=cos−1x
Solve the following for x :
`tan^(-1)((x-2)/(x-3))+tan^(-1)((x+2)/(x+3))=pi/4,|x|<1`
Evaluate the following:
`cos^-1(cos4)`
Evaluate the following:
`tan^-1(tan (6pi)/7)`
Evaluate the following:
`sec^-1(sec pi/3)`
Evaluate the following:
`cot^-1(cot pi/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)`
Evaluate the following:
`sin(tan^-1 24/7)`
Solve: `cos(sin^-1x)=1/6`
Solve the following equation for x:
cot−1x − cot−1(x + 2) =`pi/12`, x > 0
Evaluate: `cos(sin^-1 3/5+sin^-1 5/13)`
If `cos^-1 x/2+cos^-1 y/3=alpha,` then prove that `9x^2-12xy cosa+4y^2=36sin^2a.`
Find the value of the following:
`tan^-1{2cos(2sin^-1 1/2)}`
Prove that `2tan^-1(sqrt((a-b)/(a+b))tan theta/2)=cos^-1((a costheta+b)/(a+b costheta))`
Write the value of `sin^-1((-sqrt3)/2)+cos^-1((-1)/2)`
Write the difference between maximum and minimum values of sin−1 x for x ∈ [− 1, 1].
Write the value of tan−1x + tan−1 `(1/x)`for x > 0.
Evaluate sin \[\left( \tan^{- 1} \frac{3}{4} \right)\]
Write the value of cos \[\left( 2 \sin^{- 1} \frac{1}{2} \right)\]
Write the value of cos2 \[\left( \frac{1}{2} \cos^{- 1} \frac{3}{5} \right)\]
Write the value of cos−1 (cos 6).
Write the value of \[\tan^{- 1} \frac{a}{b} - \tan^{- 1} \left( \frac{a - b}{a + b} \right)\]
Show that \[\sin^{- 1} (2x\sqrt{1 - x^2}) = 2 \sin^{- 1} x\]
Write the value of \[\cos^{- 1} \left( \cos\frac{14\pi}{3} \right)\]
Write the value of \[\tan^{- 1} \left( \frac{1}{x} \right)\] for x < 0 in terms of `cot^-1x`
Find the value of \[\cos^{- 1} \left( \cos\frac{13\pi}{6} \right)\]
If x < 0, y < 0 such that xy = 1, then tan−1 x + tan−1 y equals
If \[\cos^{- 1} \frac{x}{2} + \cos^{- 1} \frac{y}{3} = \theta,\] then 9x2 − 12xy cos θ + 4y2 is equal to
If tan−1 3 + tan−1 x = tan−1 8, then x =
If \[\sin^{- 1} \left( \frac{2a}{1 - a^2} \right) + \cos^{- 1} \left( \frac{1 - a^2}{1 + a^2} \right) = \tan^{- 1} \left( \frac{2x}{1 - x^2} \right),\text{ where }a, x \in \left( 0, 1 \right)\] , then, the value of x is
Prove that : \[\cot^{- 1} \frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}} = \frac{x}{2}, 0 < x < \frac{\pi}{2}\] .
Write the value of \[\cos^{- 1} \left( - \frac{1}{2} \right) + 2 \sin^{- 1} \left( \frac{1}{2} \right)\] .
Find the value of x, if tan `[sec^(-1) (1/x) ] = sin ( tan^(-1) 2) , x > 0 `.
The value of sin `["cos"^-1 (7/25)]` is ____________.
