Advertisements
Advertisements
Question
Prove that :
`2 tan^-1 (sqrt((a-b)/(a+b))tan(x/2))=cos^-1 ((a cos x+b)/(a+b cosx))`
Solution
`2 tan^-1 (sqrt((a-b)/(a+b))tan(x/2))`
`=cos^(-1){(1-(sqrt((a-b)/(a+b))tan(x/2))^2)/(1+(sqrt((a-b)/(a+b))tan(x/2))^2)} [∵ 2 tan^(-1) (x)=cos^(−1)((1−x^2)/(1+x^2))]`
`=cos^(-1) {(1-(a-b)/(a+b)tan^2(x/2))/(1+(a-b)/(a+b)tan^2(x/2))}`
`=cos^(-1){(a+b-(a-b)tan^2(x/2))/(a+b+(a-b)tan^2(x/2))}`
`=cos^(-1){(a+b-atan^2(x/2)+btan^(x/2))/(a+b+atan^2(x/2)-btan^(x/2))}`
`=cos^(-1) {(a(1-tan^2(x/2))+b(1+tan^2(x/2)))/(a(1+tan^2(x/2))+b(1-tan^2(x/2)))}`
`=cos^(-1) {(a((1-tan^2(x/2))/(1+tan^2(x/2)))+b((1+tan^2(x/2))/(1+tan^2(x/2))))/(a((1+tan^2(x/2))/(1+tan^2(x/2)))+b((1-tan^2(x/2))/(1+tan^2(x/2))))}`
`=cos^(-1){(a((1-tan^2(x/2))/(1+tan^2(x/2)))+b)/(a+b((1-tan^2(x/2))/(1+tan^2(x/2))))}`
`=cos^(-1){(acosx+b)/(a+bcosx)}`
APPEARS IN
RELATED QUESTIONS
`sin^-1(sin pi/6)`
`sin^-1(sin (7pi)/6)`
`sin^-1(sin (17pi)/8)`
Evaluate the following:
`cos^-1(cos12)`
Write the following in the simplest form:
`tan^-1{(sqrt(1+x^2)+1)/x},x !=0`
Evaluate the following:
`cot(cos^-1 3/5)`
Prove the following result-
`tan^-1 63/16 = sin^-1 5/13 + cos^-1 3/5`
Evaluate:
`cos{sin^-1(-7/25)}`
Evaluate:
`sec{cot^-1(-5/12)}`
Evaluate:
`sin(tan^-1x+tan^-1 1/x)` for x < 0
If `cot(cos^-1 3/5+sin^-1x)=0`, find the values of x.
Find the value of `tan^-1 (x/y)-tan^-1((x-y)/(x+y))`
Solve the following equation for x:
`tan^-1 x/2+tan^-1 x/3=pi/4, 0<x<sqrt6`
Evaluate the following:
`tan{2tan^-1 1/5-pi/4}`
`2sin^-1 3/5-tan^-1 17/31=pi/4`
Prove that
`tan^-1((1-x^2)/(2x))+cot^-1((1-x^2)/(2x))=pi/2`
Prove that
`sin{tan^-1 (1-x^2)/(2x)+cos^-1 (1-x^2)/(2x)}=1`
Find the value of the following:
`tan^-1{2cos(2sin^-1 1/2)}`
Write the value of tan−1x + tan−1 `(1/x)`for x > 0.
Write the value of \[\tan^{- 1} \frac{a}{b} - \tan^{- 1} \left( \frac{a - b}{a + b} \right)\]
Write the principal value of \[\tan^{- 1} 1 + \cos^{- 1} \left( - \frac{1}{2} \right)\]
Write the principal value of \[\cos^{- 1} \left( \cos680^\circ \right)\]
Write the value of \[\cos\left( \sin^{- 1} x + \cos^{- 1} x \right), \left| x \right| \leq 1\]
Find the value of \[\tan^{- 1} \left( \tan\frac{9\pi}{8} \right)\]
The value of tan \[\left\{ \cos^{- 1} \frac{1}{5\sqrt{2}} - \sin^{- 1} \frac{4}{\sqrt{17}} \right\}\] is
The positive integral solution of the equation
\[\tan^{- 1} x + \cos^{- 1} \frac{y}{\sqrt{1 + y^2}} = \sin^{- 1} \frac{3}{\sqrt{10}}\text{ is }\]
The number of real solutions of the equation \[\sqrt{1 + \cos 2x} = \sqrt{2} \sin^{- 1} (\sin x), - \pi \leq x \leq \pi\]
It \[\tan^{- 1} \frac{x + 1}{x - 1} + \tan^{- 1} \frac{x - 1}{x} = \tan^{- 1}\] (−7), then the value of x is
tanx is periodic with period ____________.
The period of the function f(x) = tan3x is ____________.
