Advertisements
Advertisements
प्रश्न
Prove the following result:
`sin^-1 12/13+cos^-1 4/5+tan^-1 63/16=pi`
Advertisements
उत्तर
LHS = `sin^-1 12/13+cos^-1 4/5+tan^-1 63/16`
`=tan^-1 (12/13)/sqrt(1-144/169)+tan^-1 sqrt(1-16/25)/(4/5)+tan^-1 63/16` `[becausesin^-1x=tan^-1 x/sqrt(1-x^2) and cos^-1x=tan^-1 sqrt(1-x^2)/x]`
`=tan^-1 (12/13)/(5/13)+tan^-1 (3/5)/(4/5)+tan^-1 63/16`
`=tan^-1 12/5+tabn^-1 3/4+tan^-1 63/16`
`=pi+tan^-1((12/5+3/4)/(1-12/5xx3/4))+tan^-1 63/16` `[because tan^-1x+tan^-1y=pi+tan^-1((x+y)/(1-xy))]`
`=pi+tan^-1((63/20)/(-16/20))+tan^-1 63/16`
`=pi+tan^-1 (-63)/16+tan^-1 63/16`
`=pi-tan^-1 63/16+tan^-1 63/16`
= π = RHS
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`
Prove that :
`2 tan^-1 (sqrt((a-b)/(a+b))tan(x/2))=cos^-1 ((a cos x+b)/(a+b cosx))`
`sin^-1(sin (17pi)/8)`
`sin^-1(sin3)`
Evaluate the following:
`cos^-1{cos (13pi)/6}`
Evaluate the following:
`tan^-1(tan (9pi)/4)`
Evaluate the following:
`tan^-1(tan2)`
Evaluate the following:
`tan^-1(tan4)`
Evaluate the following:
`sec^-1(sec (5pi)/4)`
Evaluate the following:
`sec^-1(sec (13pi)/4)`
Evaluate the following:
`\text(cosec)^-1(\text{cosec} pi/4)`
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^-1sqrt((a-x)/(a+x)),-a<x<a`
Write the following in the simplest form:
`tan^-1(x/(a+sqrt(a^2-x^2))),-a<x<a`
Evaluate:
`cos{sin^-1(-7/25)}`
Evaluate:
`sec{cot^-1(-5/12)}`
Evaluate:
`sin(tan^-1x+tan^-1 1/x)` for x < 0
`sin(sin^-1 1/5+cos^-1x)=1`
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 equation `cos^-1 a/x-cos^-1 b/x=cos^-1 1/b-cos^-1 1/a`
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.`
Write the value of sin (cot−1 x).
Write the value of
\[\cos^{- 1} \left( \frac{1}{2} \right) + 2 \sin^{- 1} \left( \frac{1}{2} \right)\].
Write the value of sin−1 (sin 1550°).
Write the value of cos−1 (cos 350°) − sin−1 (sin 350°)
If tan−1 x + tan−1 y = `pi/4`, then write the value of x + y + xy.
Write the value of tan−1\[\left\{ \tan\left( \frac{15\pi}{4} \right) \right\}\]
Write the value of \[\sin^{- 1} \left( \frac{1}{3} \right) - \cos^{- 1} \left( - \frac{1}{3} \right)\]
If x < 0, y < 0 such that xy = 1, then write the value of tan−1 x + tan−1 y.
Write the value of \[\tan\left( 2 \tan^{- 1} \frac{1}{5} \right)\]
Write the value of \[\tan^{- 1} \left\{ 2\sin\left( 2 \cos^{- 1} \frac{\sqrt{3}}{2} \right) \right\}\]
The set of values of `\text(cosec)^-1(sqrt3/2)`
2 tan−1 {cosec (tan−1 x) − tan (cot−1 x)} is equal to
If x < 0, y < 0 such that xy = 1, then tan−1 x + tan−1 y equals
If \[\cos^{- 1} \frac{x}{2} + \cos^{- 1} \frac{y}{3} = \theta,\] then 9x2 − 12xy cos θ + 4y2 is equal to
Prove that : \[\tan^{- 1} \left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) = \frac{\pi}{4} + \frac{1}{2} \cos^{- 1} x^2 ; 1 < x < 1\].
