Advertisements
Advertisements
प्रश्न
Prove that `2tan^-1(sqrt((a-b)/(a+b))tan theta/2)=cos^-1((a costheta+b)/(a+b costheta))`
Advertisements
उत्तर
LHS = `2tan^-1(sqrt((a-b)/(a+b))tan theta/2)=cos^-1{(1-(sqrt((a-b)/(a+b))tan theta/2)^2)/(1+(sqrt((a-b)/(a+b))tan theta/2)^2)}` `[because2tan^-1(x)=cos^-1{(1-x^2)/(1+x^2)}]`
`=cos^-1{(1-(a-b)/(a+b)tan^2 theta/2)/(1+(a-b)/(a+b)tan^2 theta/2)}`
`=cos^-1{((a+b)-(a-b)tan^2 theta/2)/((a+b)+(a-b)tan^2 theta/2)}`
`=cos^-1{(a+b-atan^2 theta/2+btan^2 theta/2)/(a+b+atan^2 theta/2-btan^2 theta/2)}`
`=cos^-1{(a(1-tan^2 theta/2)+b(1+tan^2 theta/2))/(a(1+tan^2 theta/2)+b(1-tan^2 theta/2))}`
`=cos^-1{(a((1-tan^2 theta/2)/(1+tan^2 theta/2))+b((1+tan^2 theta/2)/(1+tan^2theta/2)))/(a((1+tan^2 theta/2)/(1+tan^2 theta/2))+b((1-tan^2 theta/2)/(1-tan^2 theta/2)))}` `["Dividing" N^r and D^r by 1+tan^2 theta/2]`
`=cos^-1{(a((1-tan^2 theta/2)/(1+tan^2 theta/2))+b)/(a+b((1-tan^2 theta/2)/(1-tan^2 theta/2)))}`
`=cos^-1{(acos theta+b)/(a+bcostheta)}`=RHS
APPEARS IN
संबंधित 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`
Solve the following for x :
`tan^(-1)((x-2)/(x-3))+tan^(-1)((x+2)/(x+3))=pi/4,|x|<1`
If tan-1x+tan-1y=π/4,xy<1, then write the value of x+y+xy.
Evaluate the following:
`tan^-1(tan pi/3)`
Evaluate the following:
`sec^-1(sec (13pi)/4)`
Write the following in the simplest form:
`sin^-1{(sqrt(1+x)+sqrt(1-x))/2},0<x<1`
Evaluate the following:
`sin(cos^-1 5/13)`
Evaluate the following:
`sin(tan^-1 24/7)`
Evaluate the following:
`cot(cos^-1 3/5)`
Evaluate:
`cot(sin^-1 3/4+sec^-1 4/3)`
If `cot(cos^-1 3/5+sin^-1x)=0`, find the values of x.
If `(sin^-1x)^2+(cos^-1x)^2=(17pi^2)/36,` Find x
Solve the following equation for x:
`tan^-1((1-x)/(1+x))-1/2 tan^-1x` = 0, where x > 0
Evaluate: `cos(sin^-1 3/5+sin^-1 5/13)`
`2sin^-1 3/5-tan^-1 17/31=pi/4`
`2tan^-1 3/4-tan^-1 17/31=pi/4`
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`
If x < 0, then write the value of cos−1 `((1-x^2)/(1+x^2))` in terms of tan−1 x.
Write the value of tan−1 x + tan−1 `(1/x)` for x < 0.
Write the value of sin−1
\[\left( \sin( -{600}°) \right)\].
Evaluate sin \[\left( \tan^{- 1} \frac{3}{4} \right)\]
Write the principal value of \[\cos^{- 1} \left( \cos\frac{2\pi}{3} \right) + \sin^{- 1} \left( \sin\frac{2\pi}{3} \right)\]
Write the principal value of \[\cos^{- 1} \left( \cos680^\circ \right)\]
The value of tan \[\left\{ \cos^{- 1} \frac{1}{5\sqrt{2}} - \sin^{- 1} \frac{4}{\sqrt{17}} \right\}\] is
If sin−1 x − cos−1 x = `pi/6` , then x =
sin\[\left[ \cot^{- 1} \left\{ \tan\left( \cos^{- 1} x \right) \right\} \right]\] is equal to
The number of solutions of the equation \[\tan^{- 1} 2x + \tan^{- 1} 3x = \frac{\pi}{4}\] is
If α = \[\tan^{- 1} \left( \tan\frac{5\pi}{4} \right) \text{ and }\beta = \tan^{- 1} \left( - \tan\frac{2\pi}{3} \right)\] , then
The value of \[\sin^{- 1} \left( \cos\frac{33\pi}{5} \right)\] is
The value of \[\cos^{- 1} \left( \cos\frac{5\pi}{3} \right) + \sin^{- 1} \left( \sin\frac{5\pi}{3} \right)\] is
The value of \[\sin\left( 2\left( \tan^{- 1} 0 . 75 \right) \right)\] is equal to
Find : \[\int\frac{2 \cos x}{\left( 1 - \sin x \right) \left( 1 + \sin^2 x \right)}dx\] .
If \[\tan^{- 1} \left( \frac{1}{1 + 1 . 2} \right) + \tan^{- 1} \left( \frac{1}{1 + 2 . 3} \right) + . . . + \tan^{- 1} \left( \frac{1}{1 + n . \left( n + 1 \right)} \right) = \tan^{- 1} \theta\] , then find the value of θ.
Find the domain of `sec^(-1)(3x-1)`.
Find the simplified form of `cos^-1 (3/5 cosx + 4/5 sin x)`, where x ∈ `[(-3pi)/4, pi/4]`
Find the value of `sin^-1(cos((33π)/5))`.
