Question

Assertion :${\sec }^{-1}5<{\tan }^{-1}5<{\tan }^{-1}7$ Reason: ${\sec }^{-1}x<{\tan }^{-1}x$ if $x\ge 1$ , ${\sec }^{-1}x>{\tan }^{-1}x$ is $x\le -1$ and ${\tan }^{-1}{x}_1>{\tan }^{-1}{x}_2$ if $x_1>x_2$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

${\tan }^{-1}x$ is an increasing function by definition

${\sec }^{-1}x={\tan }^{-1}\sqrt{x^2-1}$ for $x\ge 1$

$\therefore {\sec }^{-1}x<{\tan }^{-1}x$

$\Rightarrow {\tan }^{-1}\sqrt{x^2-1}<{\tan }^{-1}x$

$\Rightarrow \sqrt{x^2-1}<x\Rightarrow x^2-1<x^2$ which is true

Also for $x\le -1;$

${\sec }^{-1}x\in (\frac{\pi }{2},\pi ]$ and

${\tan }^{-1}x\in (-\frac{\pi }{2},-\frac{\pi }{4}]$

$\Rightarrow {\sec }^{-1}x>{\tan }^{-1}x$

Therefore reason is true and assertion is explained by reason.