Advertisements
Advertisements
Question
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.`
Advertisements
Solution
We know
`tan^-1x+tan^-1y=tan^-1((x+y)/(1-xy)),xy>1`
`thereforetan^-1 (2ab)/(a^2-b^2)+tan^-1 (2xy)/(x^2-y^2)=tan^-1(((2ab)/(a^2-b^2)+(2xy)/(x^2-y^2))/(1-(2ab)/(a^2-b^2) (2xy)/(x^2-y^2)))`
`=tan^-1(((2(abx^2-aby^2+xya^2-xyb^2))/((a^2-b^2)(x^2-y^2)))/((a^2x^2-a^2y^2-x^2b^2+y^2b^2-4abxy)/((a^2-b^2)(x^2-y^2))))`
`=tan^-1((2(abx^2-aby^2+xya^2-xyb^2))/(a^2x^2-a^2y^2-x^2b^2+y^2b^2-2abxy))`
`=tan^-1((2(ax-by)(ay+bx))/((ax-by)^2-(ay+bx)^2))`
`=tan^-1((2alphabeta)/(alpha^2-beta^2))` `[because alpha=ax-by and beta = ay+bx]`
APPEARS IN
RELATED QUESTIONS
Solve the equation for x:sin−1x+sin−1(1−x)=cos−1x
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`
Solve the following for x:
`sin^(-1)(1-x)-2sin^-1 x=pi/2`
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.
Find the principal values of the following:
`cos^-1(tan (3pi)/4)`
`sin^-1(sin pi/6)`
Evaluate the following:
`cos^-1{cos (5pi)/4}`
Evaluate the following:
`cos^-1(cos5)`
Evaluate the following:
`sec^-1(sec (2pi)/3)`
Write the following in the simplest form:
`sin^-1{(x+sqrt(1-x^2))/sqrt2},-1<x<1`
Evaluate the following:
`cot(cos^-1 3/5)`
Solve: `cos(sin^-1x)=1/6`
Evaluate:
`cos{sin^-1(-7/25)}`
Evaluate:
`cot(sin^-1 3/4+sec^-1 4/3)`
Evaluate:
`sin(tan^-1x+tan^-1 1/x)` for x < 0
If `(sin^-1x)^2+(cos^-1x)^2=(17pi^2)/36,` Find x
`sin(sin^-1 1/5+cos^-1x)=1`
`4sin^-1x=pi-cos^-1x`
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(2+x)+tan^-1(2-x)=tan^-1 2/3, where x< -sqrt3 or, x>sqrt3`
If `cos^-1 x/2+cos^-1 y/3=alpha,` then prove that `9x^2-12xy cosa+4y^2=36sin^2a.`
Solve `cos^-1sqrt3x+cos^-1x=pi/2`
Find the value of the following:
`cos(sec^-1x+\text(cosec)^-1x),` | x | ≥ 1
Solve the following equation for x:
`tan^-1((x-2)/(x-1))+tan^-1((x+2)/(x+1))=pi/4`
Evaluate sin
\[\left( \frac{1}{2} \cos^{- 1} \frac{4}{5} \right)\]
Write the value of cos−1 \[\left( \tan\frac{3\pi}{4} \right)\]
Write the value of cos \[\left( 2 \sin^{- 1} \frac{1}{2} \right)\]
Write the value of cos−1 (cos 350°) − sin−1 (sin 350°)
Show that \[\sin^{- 1} (2x\sqrt{1 - x^2}) = 2 \sin^{- 1} x\]
Write the principal value of \[\cos^{- 1} \left( \cos\frac{2\pi}{3} \right) + \sin^{- 1} \left( \sin\frac{2\pi}{3} \right)\]
Write the principal value of \[\cos^{- 1} \left( \cos680^\circ \right)\]
Write the value of \[\cos\left( \sin^{- 1} x + \cos^{- 1} x \right), \left| x \right| \leq 1\]
The value of tan \[\left\{ \cos^{- 1} \frac{1}{5\sqrt{2}} - \sin^{- 1} \frac{4}{\sqrt{17}} \right\}\] is
\[\text{ If } u = \cot^{- 1} \sqrt{\tan \theta} - \tan^{- 1} \sqrt{\tan \theta}\text{ then }, \tan\left( \frac{\pi}{4} - \frac{u}{2} \right) =\]
If \[\cos^{- 1} \frac{x}{2} + \cos^{- 1} \frac{y}{3} = \theta,\] then 9x2 − 12xy cos θ + 4y2 is equal to
The value of \[\cos^{- 1} \left( \cos\frac{5\pi}{3} \right) + \sin^{- 1} \left( \sin\frac{5\pi}{3} \right)\] is
If \[\tan^{- 1} \left( \frac{1}{1 + 1 . 2} \right) + \tan^{- 1} \left( \frac{1}{1 + 2 . 3} \right) + . . . + \tan^{- 1} \left( \frac{1}{1 + n . \left( n + 1 \right)} \right) = \tan^{- 1} \theta\] , then find the value of θ.
The value of sin `["cos"^-1 (7/25)]` is ____________.
