Question

Assertion :STATEMENT 1: Domain of ${\tan }^{-1}x$ and ${\cot }^{-1}x$ is $R$ Reason: STATEMENT 2: $f\left(x\right)=\tan x$ and $g\left(x\right)=\cot x$ are unbounded function

• A. Both the statements are TRUE and STATEMENT 2 is the correct explanation of STATEMENT 1
• B. Both the statements are TRUE and STATEMENT 2 is NOT the correct explanation of STATEMENT 1
• C. STATEMENT 1 is TRUE and STATEMENT 2 is FALSE
• D. STATEMENT 1 is FALSE and STATEMENT 2 is TRUE

Answer 1

The correct option is B Both the statements are TRUE and STATEMENT 2 is NOT the correct explanation of STATEMENT 1

$f\left(x\right)=tanx$

Then for all $xϵ(-\infty ,\infty )$ $f\left(x\right)ϵ(-\infty ,\infty )$.

The same is applicable for $f\left(x\right)=cotx$.

That is why, the domain of $ta{n}^{-1}\left(x\right)$ and $co{t}^{-1}\left(x\right)$ is R.

Also, the range of $f\left(x\right)=tanx$ and $f\left(x\right)=cot\left(x\right)$ indicate that, they are unbounded function, unlike $sin\left(x\right)$ or $cos\left(x\right)$ which are bounded functions, whith range of values between -1 and 1 only.

Thus the reason is correct but is not the correct explanation for the assertion.