Advertisements
Advertisements
Question
Write the value of \[\tan^{- 1} \frac{a}{b} - \tan^{- 1} \left( \frac{a - b}{a + b} \right)\]
Advertisements
Solution
We know that
\[\tan^{- 1} x - \tan^{- 1} y = \tan^{- 1} \left( \frac{x - y}{1 + xy} \right)\]
Now,
\[\tan^{- 1} \frac{a}{b} - \tan^{- 1} \left( \frac{a - b}{a + b} \right) = \tan^{- 1} \left( \frac{\frac{a}{b} - \frac{a - b}{a + b}}{1 + \frac{a}{b}\frac{a - b}{a + b}} \right)\]
\[ = \tan^{- 1} \left( \frac{\frac{a^2 + ab - ab + b^2}{b\left( a + b \right)}}{\frac{ab + b^2 - ab + a^2}{b\left( a + b \right)}} \right)\]
\[ = \tan^{- 1} \left( 1 \right)\]
\[ = \tan^{- 1} \left( \tan\frac{\pi}{4} \right) \left[ \because \tan\frac{\pi}{4} = 1 \right]\]
\[ = \frac{\pi}{4}\]
∴ \[\tan^{- 1} \frac{a}{b} - \tan^{- 1} \left( \frac{a - b}{a + b} \right) = \frac{\pi}{4}\]
APPEARS IN
RELATED QUESTIONS
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 for x:
`2tan^(-1)(cosx)=tan^(-1)(2"cosec" x)`
If tan-1x+tan-1y=π/4,xy<1, then write the value of x+y+xy.
Find the principal values of the following:
`cos^-1(-1/sqrt2)`
Find the principal values of the following:
`cos^-1(sin (4pi)/3)`
`sin^-1(sin pi/6)`
Evaluate the following:
`cos^-1{cos (5pi)/4}`
Evaluate the following:
`cos^-1{cos (13pi)/6}`
Evaluate the following:
`cos^-1(cos5)`
Write the following in the simplest form:
`tan^-1{x+sqrt(1+x^2)},x in R `
Write the following in the simplest form:
`tan^-1{sqrt(1+x^2)-x},x in R`
Write the following in the simplest form:
`tan^-1{(sqrt(1+x^2)+1)/x},x !=0`
Evaluate:
`sin(tan^-1x+tan^-1 1/x)` for x < 0
Prove the following result:
`sin^-1 12/13+cos^-1 4/5+tan^-1 63/16=pi`
Solve `cos^-1sqrt3x+cos^-1x=pi/2`
Evaluate the following:
`sin(2tan^-1 2/3)+cos(tan^-1sqrt3)`
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)`
Show that `2tan^-1x+sin^-1 (2x)/(1+x^2)` is constant for x ≥ 1, find that constant.
Solve the following equation for x:
`3sin^-1 (2x)/(1+x^2)-4cos^-1 (1-x^2)/(1+x^2)+2tan^-1 (2x)/(1-x^2)=pi/3`
Solve the following equation for x:
`tan^-1((x-2)/(x-1))+tan^-1((x+2)/(x+1))=pi/4`
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`
Write the value of tan−1x + tan−1 `(1/x)`for x > 0.
Write the value of cos\[\left( 2 \sin^{- 1} \frac{1}{3} \right)\]
Evaluate sin
\[\left( \frac{1}{2} \cos^{- 1} \frac{4}{5} \right)\]
Evaluate sin \[\left( \tan^{- 1} \frac{3}{4} \right)\]
Write the value of sin \[\left\{ \frac{\pi}{3} - \sin^{- 1} \left( - \frac{1}{2} \right) \right\}\]
Show that \[\sin^{- 1} (2x\sqrt{1 - x^2}) = 2 \sin^{- 1} x\]
Evaluate: \[\sin^{- 1} \left( \sin\frac{3\pi}{5} \right)\]
Write the value of \[\sin^{- 1} \left( \frac{1}{3} \right) - \cos^{- 1} \left( - \frac{1}{3} \right)\]
If 4 sin−1 x + cos−1 x = π, then what is the value of x?
The set of values of `\text(cosec)^-1(sqrt3/2)`
If \[\tan^{- 1} \left( \frac{\sqrt{1 + x^2} - \sqrt{1 - x^2}}{\sqrt{1 + x^2} + \sqrt{1 - x^2}} \right)\] = α, then x2 =
2 tan−1 {cosec (tan−1 x) − tan (cot−1 x)} is equal to
The number of real solutions of the equation \[\sqrt{1 + \cos 2x} = \sqrt{2} \sin^{- 1} (\sin x), - \pi \leq x \leq \pi\]
\[\tan^{- 1} \frac{1}{11} + \tan^{- 1} \frac{2}{11}\] is equal to
The value of \[\sin^{- 1} \left( \cos\frac{33\pi}{5} \right)\] is
If θ = sin−1 {sin (−600°)}, then one of the possible values of θ is
If x = a (2θ – sin 2θ) and y = a (1 – cos 2θ), find \[\frac{dy}{dx}\] When \[\theta = \frac{\pi}{3}\] .
Find the simplified form of `cos^-1 (3/5 cosx + 4/5 sin x)`, where x ∈ `[(-3pi)/4, pi/4]`
