Advertisements
Advertisements
प्रश्न
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`
उत्तर
We have to find the value of `tan(1/2)[sin^(-1)((2x)/(1+x^2))+cos^(-1)((1-y^2)/(1+y^2))]`
We know that: `sin^(-1)(2x)/(1+x^2)=2tan^(-1)x for |x| ≤ 1 …… (1)`
`cos^(-1)(1-y^2)/(1+y^2)=2tan^(-1)y for y > 0 …… (2)`
`Now sin^(-1)((2x)/(1+x^2)) + cos^(-1)((1-y^2)/(1+y^2))=2tan^(-1)x+2tan^(-1)y`
`tan(1/2)[sin^(-1)((2x)/(1+x^2))+cos^(-1)((1-y^2)/(1+y^2))]=tan(1/2)(2tan^(-1)x+2tan^(-1)y)=tan(tan^(-1)x+tan^(-1)y)`
Since, ` tan^(−1)x + tan^(−1)y = tan^(−1)((x+y)/(1-xy)) for xy < 1`
`therefore tan(1/2)[sin^(-1)((2x)/(1+x^2))+cos^(-1)((1-y^2)/(1+y^2))]=tan(tan^(−1)((x+y)/(1-xy)))=(x+y)/(1-xy)`
APPEARS IN
संबंधित प्रश्न
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`
Find the principal values of the following:
`cos^-1(-1/sqrt2)`
`sin^-1(sin pi/6)`
`sin^-1(sin (7pi)/6)`
`sin^-1(sin (17pi)/8)`
Evaluate the following:
`cos^-1{cos(-pi/4)}`
Evaluate the following:
`cos^-1{cos (13pi)/6}`
Evaluate the following:
`cot(cos^-1 3/5)`
Evaluate the following:
`cos(tan^-1 24/7)`
Evaluate:
`sin(tan^-1x+tan^-1 1/x)` for x < 0
Prove the following result:
`tan^-1 1/7+tan^-1 1/13=tan^-1 2/9`
If `cos^-1 x/2+cos^-1 y/3=alpha,` then prove that `9x^2-12xy cosa+4y^2=36sin^2a.`
Show that `2tan^-1x+sin^-1 (2x)/(1+x^2)` is constant for x ≥ 1, find that constant.
Solve the following equation for x:
`tan^-1((2x)/(1-x^2))+cot^-1((1-x^2)/(2x))=(2pi)/3,x>0`
Prove that `2tan^-1(sqrt((a-b)/(a+b))tan theta/2)=cos^-1((a costheta+b)/(a+b costheta))`
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−1\[\left\{ \tan\left( \frac{15\pi}{4} \right) \right\}\]
If \[\tan^{- 1} (\sqrt{3}) + \cot^{- 1} x = \frac{\pi}{2},\] find x.
Write the value of \[\cos^{- 1} \left( \cos\frac{14\pi}{3} \right)\]
Write the principal value of \[\sin^{- 1} \left\{ \cos\left( \sin^{- 1} \frac{1}{2} \right) \right\}\]
Write the value of `cot^-1(-x)` for all `x in R` in terms of `cot^-1(x)`
If \[\cos\left( \tan^{- 1} x + \cot^{- 1} \sqrt{3} \right) = 0\] , find the value of x.
The number of real solutions of the equation \[\sqrt{1 + \cos 2x} = \sqrt{2} \sin^{- 1} (\sin x), - \pi \leq x \leq \pi\]
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
If 2 tan−1 (cos θ) = tan−1 (2 cosec θ), (θ ≠ 0), then find the value of θ.
The equation sin-1 x – cos-1 x = cos-1 `(sqrt3/2)` has ____________.
