Advertisements
Advertisements
प्रश्न
Write the value of sin (cot−1 x).
Advertisements
उत्तर
We know
\[\cot^{- 1} x = \tan^{- 1} \frac{1}{x}\]
Now, we have
\[\sin\left( \cot^{- 1} x \right) = \sin\left( \tan^{- 1} \frac{1}{x} \right)\]
\[ = \sin\left[ \sin^{- 1} \left( \frac{\frac{1}{x}}{\sqrt{1 + \frac{1}{x^2}}} \right) \right] \left[ \because \tan^{- 1} x = \sin^{- 1} \left( \frac{x}{\sqrt{1 + x^2}} \right) \right]\]
\[ = \sin\left[ \sin^{- 1} \left( \frac{\frac{1}{x}}{\frac{\sqrt{x^2 + 1}}{x}} \right) \right]\]
\[ = \sin\left( \sin^{- 1} \frac{1}{\sqrt{x^2 + 1}} \right)\]
\[ = \frac{1}{\sqrt{x^2 + 1}} \left[ \because \sin\left( \sin^{- 1} x = x \right) \right]\]
Hence,
\[\sin\left( \cot^{- 1} x \right) = \frac{1}{\sqrt{x^2 - 1}}\]
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`
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.
If `(sin^-1x)^2 + (sin^-1y)^2+(sin^-1z)^2=3/4pi^2,` find the value of x2 + y2 + z2
`sin^-1(sin pi/6)`
`sin^-1(sin (7pi)/6)`
`sin^-1(sin3)`
`sin^-1(sin4)`
Evaluate the following:
`cos^-1(cos4)`
Evaluate the following:
`cos^-1(cos12)`
Evaluate the following:
`tan^-1(tan (7pi)/6)`
Evaluate the following:
`tan^-1(tan1)`
Evaluate the following:
`tan^-1(tan2)`
Evaluate the following:
`sec^-1(sec (5pi)/4)`
Evaluate the following:
`\text(cosec)^-1(\text{cosec} pi/4)`
Evaluate the following:
`cot^-1(cot (19pi)/6)`
Write the following in the simplest form:
`tan^-1(x/(a+sqrt(a^2-x^2))),-a<x<a`
Prove the following result
`sin(cos^-1 3/5+sin^-1 5/13)=63/65`
Evaluate:
`cot{sec^-1(-13/5)}`
Evaluate:
`cos(tan^-1 3/4)`
Evaluate: `sin{cos^-1(-3/5)+cot^-1(-5/12)}`
Evaluate:
`sin(tan^-1x+tan^-1 1/x)` for x > 0
If `(sin^-1x)^2+(cos^-1x)^2=(17pi^2)/36,` Find x
`sin^-1x=pi/6+cos^-1x`
Prove the following result:
`tan^-1 1/4+tan^-1 2/9=sin^-1 1/sqrt5`
Solve the following equation for x:
tan−1(x −1) + tan−1x tan−1(x + 1) = tan−13x
Solve the following equation for x:
`tan^-1((x-2)/(x-4))+tan^-1((x+2)/(x+4))=pi/4`
Solve the following equation for x:
`tan^-1 (x-2)/(x-1)+tan^-1 (x+2)/(x+1)=pi/4`
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)`
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.`
If `sin^-1x+sin^-1y+sin^-1z=(3pi)/2,` then write the value of x + y + z.
If x > 1, then write the value of sin−1 `((2x)/(1+x^2))` in terms of tan−1 x.
Write the value of cos−1 \[\left( \cos\frac{5\pi}{4} \right)\]
Write the principal value of `sin^-1(-1/2)`
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`
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 =
If α = \[\tan^{- 1} \left( \frac{\sqrt{3}x}{2y - x} \right), \beta = \tan^{- 1} \left( \frac{2x - y}{\sqrt{3}y} \right),\]
then α − β =
The value of \[\sin^{- 1} \left( \cos\frac{33\pi}{5} \right)\] is
The domain of \[\cos^{- 1} \left( x^2 - 4 \right)\] is
If 2 tan−1 (cos θ) = tan−1 (2 cosec θ), (θ ≠ 0), then find the value of θ.
