Question

Assertion(A): ${\cos }^{-1}x$ and ${\tan }^{-1}x$ are positive for all positive real values of $x$ in their domain.
Reason(R): The domain of $f\left(x\right)={\cos }^{-1}x+{\tan }^{-1}x$ is $[-1,1].$

• A. Both A and R are true and R is the correct explanation of A
• B. Both A and R are true but R is not correct explanation of A
• C. A is true but R is false
• D. A is false but R is true

Answer 1

Answer: (B)

Both A and R are true but R is not correct explanation of A

Assertion :

${\cos }^{-1}x$ range is $[0,\pi ]$ and domain is $[-1,1]$
$\therefore {\cos }^{-1}x>0$ for all $x>0$

${\tan }^{-1}x$ range is $(\frac{-\pi }{2},\frac{\pi }{2})$ and range is $R$

$\forall x>0⟹{\tan }^{-1}x>0$

Reason :

The domain of ${\cos }^{-1}x$ is $[-1,1]$

The domain of ${\tan }^{-1}x$ is $R$

So The domain of ${\cos }^{-1}x+{\tan }^{-1}x$ is $R\cap [-1,1]$

$⟹[-1,1]$