Considering only the principal values of inverse functions, the set A = x ge 0: tan-1 (2x) + tan-1 (3x) = (π/4)

Question

Considering only the principal values of inverse functions, the set A = x ge 0: tan-1 (2x) + tan-1 (3x) = (π/4)  (A) is an empty set (B) Contains more than two elements (C) Contains two elements (D) is a singleton

Tardigrade Admin · Accepted Answer

tan-1(2x)+ tan-1(3x)=π/4 ⇒ (5x/1-6x2) = 1 ⇒ 6x2+5x-1=0 x=-1 or x=(1/6) x = (1/6) ∵ x >0

Q. Considering only the principal values of inverse functions, the set $A = {x \geq 0 : tan^{- 1} (2 x) + tan^{- 1} (3 x) = \frac{π}{4}}$

is an empty set

Contains more than two elements

Contains two elements

is a singleton

$tan^{- 1} (2 x) + tan^{- 1} (3 x) = π /4$
$\Rightarrow \frac{5 x}{1 - 6 x ^{2}} = 1$
$\Rightarrow 6 x^{2} + 5 x - 1 = 0$
$x = - 1$ or $x = \frac{1}{6}$
$x = \frac{1}{6} ∵ x > 0$

Q. Considering only the principal values of inverse functions, the set A={x≥0:tan−1(2x)+tan−1(3x)=4π​}