Exam

NTA Abhyas

Year

2020

Subject

Mathematics

Chapter

Inverse Trigonometric Functions

NTA Abhyas 2020 · Mathematics · Inverse Trigonometric Functions

If ( ^ - 1 (1 - x)) = ( ^ - 1 (- x)) , then x is

If $\sin \left(\cot^{- 1} \left(1 - x\right)\right) = \cos \left(\tan^{- 1} \left(- x\right)\right)$, then $x$ is

Step-by-step solution

$\sin \left(\cot^{- 1} \left(1 - x\right)\right) = \cos \left(\tan^{- 1} \left(- x\right)\right)$
Let, $\cot^{- 1} \left(1 - x\right) = \theta$
$⇒ cot \theta = \left(1 - x\right)$

$⇒ sin \theta = \dfrac{1}{\sqrt{2 - 2x + {x}^{2}}}$
Let, $\tan^{- 1} \left(- x\right) = ϕ$
$⇒ tan ϕ = - x ⇒ cos ϕ = \dfrac{1}{\sqrt{1 + {x}^{2}}}$

$∴ sin \theta = cos ϕ$
$⇒ \dfrac{1}{\sqrt{2 - 2x + {x}^{2}}} = \dfrac{1}{\sqrt{1 + {x}^{2}}}$
$⇒ 1 + {x}^{2} = 2 - 2x + {x}^{2}$
$⇒ x = \dfrac{1}{2}$