Find maximum value of x for which 2 tan -1 x-cos -11-x2-1-x2 is independent of x-

The correct option is A 0Let x = tan θ, π/2 lt; θ lt; π/2 2 θ +cos^ 1 cos 2 θ (i)0 ≤ 2 θ lt; π/2 cos^ 1cos 2 θ =2 θ So given expression =4 θ =4 ta ...

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Standard XII

Mathematics

Domain and Range of Basic Inverse Trigonometric Functions

Find maximum ...

Question

Find maximum value of x for which $2 t a n^{- 1} x + c o s^{- 1} (\frac{1 - x^{2}}{1 + x^{2}})$ is independent of x.

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Solution

Let $x = t a n θ, - \frac{π}{2} < θ < \frac{π}{2} 2 θ + c o s^{- 1} c o s 2 θ$

$(i) 0 \leq 2 θ < \frac{π}{2} c o s^{- 1} c o s 2 θ = 2 θ$
So given expression = $4 θ = 4 t a n^{- 1} x$
$(i i) - \frac{π}{2} < θ \leq 0 \Rightarrow - π < 2 θ \leq 0 c o s^{- 1} c o s 2 θ = - 2 θ$
So, given expression becomes independent of $θ$
For $- \frac{π}{2} < θ \leq 0 \Rightarrow - \infty < x \leq 0$

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Similar questions

Q. Find maximum value of x for which

2 t a n^{- 1} x + c o s^{- 1} (\frac{1 - x^{2}}{1 + x^{2}})

is independent of x.

Q. Value of

x

for which

{cos}^{- 1} (\frac{1 - x^{2}}{1 + x^{2}}) = 2 {tan}^{- 1} x

satisfied is

x ϵ [a, \infty)

.
Find the value of

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Q. Write the values of

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for which

2 {tan}^{- 1} x = {cos}^{- 1} \frac{1 - x^{2}}{1 + x^{2}}

, holds.

Q. Prove the following

(1) {sin}^{- 1} (\frac{2 x}{1 + x^{2}}) = 2 {tan}^{- 1} x, | x | \leq 1

(2) {cos}^{- 1} (\frac{1 - x^{2}}{1 + x^{2}}) = 2 {tan}^{- 1} x, x \geq 0

(3) {tan}^{- 1} (\frac{2 x}{1 - x^{2}}) = 2 {tan}^{- 1} x, - 1 < x < 1

y = c o s^{- 1} (\frac{1 - x^{2}}{1 + x^{2}}); 0 < x < 1

x = t a n θ

y = c o s^{- 1} (c o s 2 θ) = 2 θ

y = 2 t a n^{- 1} x

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Find maximum value of x for which 2 tan−1x+cos−1(1−x21+x2) is independent of x.

Let x=tan θ,−π2<θ<π22 θ+cos−1cos 2θ (i)0≤2θ<π2cos−1cos 2θ=2θ So given expression =4θ=4tan−1x (ii)−π2<θ≤0⇒−π<2θ≤0cos−1cos2θ=−2θ So, given expression becomes independent of θ For −π2<θ≤0⇒−∞<x≤0

Find maximum value of x for which $2 t a n^{- 1} x + c o s^{- 1} (\frac{1 - x^{2}}{1 + x^{2}})$ is independent of x.