Find maximum value of x for which 2 t a n − 1 x + c o s − 1 ( 1 − x 2 1 + x 2 ) is independent of x. Options: A . &nbsp 0 B . &nbsp 2 C . &nbsp 3 D . &nbsp 1

Answer: Option A : A Let x = t a n θ , − π 2 < θ < π 2 2 θ + c o s − 1 c o s 2 θ ( i ) 0 ≤ 2 θ < π 2 .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 ) − π 2 < θ ≤ 0 ⇒ − π < 2 θ ≤ 0 c o s − 1 c o s 2 θ = − 2 θ So, given expression becomes independent of θ For − π 2 < θ ≤ 0 ⇒ − ∞ < x ≤ 0

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

Discussion Forum : Inverse Trigonometric Functions

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.

Options:

A . 0

B . 2

C . 3

D . 1

Answer: Option A
:
A
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} .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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Discussion Forum : Inverse Trigonometric Functions

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