For what interval of variation of x, the identity arc cos (1 - x2/1 + x2) = - 2 arc tan x is true ?

Question

For what interval of variation of x, the identity arc cos (1 - x2/1 + x2) = - 2 arc tan x is true ? (A) 0 ≤ x < ∞  (B) - ∞ < x le 0  (C) 1 < x < ∞  (D) 0 le x le 1

Tardigrade Admin · Accepted Answer

Given, arc cos (1-x2/1+x2)=-2 ⋅ arc tan x We know that, arc cos (1-x2/1+x2)=2 arc tan x is true for 0 ≤ x<∞ ∴ arc cos (1-x2/1+x2)=-2 arc tan x is true for -∞< x ≤ 0

Q. For what interval of variation of $x$ , the identity $a rc cos \frac{1 - x ^{2}}{1 + x ^{2}} = - 2 a rc tan x$ is true ?

$0 \leq x < \infty$

$- \infty < x \leq 0$

$1 < x < \infty$

$0 \leq x \leq 1$

Given,
arc cos $\frac{1 - x ^{2}}{1 + x ^{2}} = - 2 \cdot$ arc tan x
We know that,
arc cos $\frac{1 - x ^{2}}{1 + x ^{2}} = 2$ arc $tan x$ is true for $0 \leq x < \infty$
$∴$ arc cos $\frac{1 - x ^{2}}{1 + x ^{2}} = - 2$ arc tan x is true
for $- \infty < x \leq 0$

Q. For what interval of variation of x, the identity arccos1+x21−x2​=−2arctanx is true ?

0≤x<∞

−∞<x≤0

1<x<∞

0≤x≤1