Question

Assertion :If $0<\alpha <\beta <\frac{\pi }{2}$ then $\frac{\tan \beta }{\tan \alpha }>\frac{\alpha }{\beta }$ Reason: $x\tan x$ is increasing for $0<x<\frac{\pi }{2}$

• 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

Let $f\left(x\right)=x\tan x$

$\Rightarrow f^{\prime }\left(x\right)=\tan x.1+x{\sec }^{2}x>0,$ for $x\in (0,\frac{\pi }{2})$

So, $f\left(x\right)$ is increasing for $x\in (0,\frac{\pi }{2})$

Since $0<\alpha <\beta <\frac{\pi }{2}$

$\therefore f\left(\beta \right)>f\left(\alpha \right)$

$\Rightarrow \beta \tan \beta >\alpha \tan \alpha \Rightarrow \frac{\tan \beta }{\tan \alpha }>\frac{\alpha }{\beta }$