Advertisements
Advertisements
प्रश्न
If −1 < x < 0, then write the value of `sin^-1((2x)/(1+x^2))+cos^-1((1-x^2)/(1+x^2))`
Advertisements
उत्तर
Let `x=-tany`
Where `0<y< pi/2`
Then,
`sin^-1((2x)/(1+x^2))+cos^-1((1-x^2)/(1+x^2))=sin^-1((-2tany)/(1+tan^2y))+cos^-1((1-tan^2y)/(1+tan^2y))`
`=sin^-1{-sin(2y)}+cos^-1{cos(2y)}`
`=-sin^-1{sin(2y)}+cos^-1{cos(2y)}`
`=-2y+2y`
= 0
`therefore sin^-1((2x)/(1+x^2))+cos^-1((1-x^2)/(1+x^2))=0`
APPEARS IN
संबंधित प्रश्न
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 :
`tan^(-1)((x-2)/(x-3))+tan^(-1)((x+2)/(x+3))=pi/4,|x|<1`
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`
If `(sin^-1x)^2 + (sin^-1y)^2+(sin^-1z)^2=3/4pi^2,` find the value of x2 + y2 + z2
Evaluate the following:
`sec^-1(sec (5pi)/4)`
Evaluate the following:
`\text(cosec)^-1(\text{cosec} pi/4)`
Evaluate the following:
`cosec^-1{cosec (-(9pi)/4)}`
Evaluate the following:
`cot^-1(cot pi/3)`
Write the following in the simplest form:
`tan^-1{(sqrt(1+x^2)+1)/x},x !=0`
Evaluate the following:
`sin(cos^-1 5/13)`
Prove the following result
`tan(cos^-1 4/5+tan^-1 2/3)=17/6`
Prove the following result
`sin(cos^-1 3/5+sin^-1 5/13)=63/65`
Evaluate:
`cos{sin^-1(-7/25)}`
Evaluate:
`sin(tan^-1x+tan^-1 1/x)` for x > 0
If `sin^-1x+sin^-1y=pi/3` and `cos^-1x-cos^-1y=pi/6`, find the values of x and y.
`sin^-1x=pi/6+cos^-1x`
`tan^-1x+2cot^-1x=(2x)/3`
Solve the following equation for x:
tan−1(x + 2) + tan−1(x − 2) = tan−1 `(8/79)`, x > 0
Evaluate: `cos(sin^-1 3/5+sin^-1 5/13)`
Solve the following:
`cos^-1x+sin^-1 x/2=π/6`
`tan^-1 1/4+tan^-1 2/9=1/2cos^-1 3/2=1/2sin^-1(4/5)`
`sin^-1 4/5+2tan^-1 1/3=pi/2`
Find the value of the following:
`cos(sec^-1x+\text(cosec)^-1x),` | x | ≥ 1
Solve the following equation for x:
`2tan^-1(sinx)=tan^-1(2sinx),x!=pi/2`
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 sin (cot−1 x).
Write the value of cos−1 (cos 350°) − sin−1 (sin 350°)
If 4 sin−1 x + cos−1 x = π, then what is the value of x?
Write the value of \[\tan\left( 2 \tan^{- 1} \frac{1}{5} \right)\]
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 \[\sqrt{1 + \cos 2x} = \sqrt{2} \sin^{- 1} (\sin x), - \pi \leq x \leq \pi\]
\[\tan^{- 1} \frac{1}{11} + \tan^{- 1} \frac{2}{11}\] is equal to
If \[\cos^{- 1} x > \sin^{- 1} x\], then
The value of sin \[\left( \frac{1}{4} \sin^{- 1} \frac{\sqrt{63}}{8} \right)\] is
\[\cot\left( \frac{\pi}{4} - 2 \cot^{- 1} 3 \right) =\]
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
If x > 1, then \[2 \tan^{- 1} x + \sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\] is equal to
If x = a (2θ – sin 2θ) and y = a (1 – cos 2θ), find \[\frac{dy}{dx}\] When \[\theta = \frac{\pi}{3}\] .
Write the value of \[\cos^{- 1} \left( - \frac{1}{2} \right) + 2 \sin^{- 1} \left( \frac{1}{2} \right)\] .
Find the domain of `sec^(-1) x-tan^(-1)x`
