Question

Assertion $\left(A\right):$ ${\cos }^{-1}x$ and ${\tan }^{-1}x$ are positive for all positive real values of $x$ in their domain.

Reason $\left(R\right):$ 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

The correct option is B Both $A$ and $R$ are true but $R$ is not correct explanation of $A$

Assertion:
For ${\cos }^{-1}x,$ range is $[0,\pi ]$ and domain is $[-1,1].$
$\therefore {\cos }^{-1}x>0$ for all $x$ in its domain

For ${\tan }^{-1}x,$ range is $(-\frac{\pi }{2},\frac{\pi }{2})$ and domain is $R.$
$\forall x>0\Rightarrow {\tan }^{-1}x>0$

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 $R\cap [-1,1]$ i.e., $[-1,1]$
So, it's also true but not the correct explanation of $A$ as $R$ doesn't give any information about range.