Question

If $a+\frac{\pi }{2}<2{\tan }^{-1}x+3{\cot }^{-1}x<b$ then ${}^{\prime }{a}^{\prime }$ and${}^{\prime }{b}^{\prime }$ are respectively.

• A. $0$ and $2\pi$
• B. $0$ and $\pi$
• C. $\frac{-\pi }{2}$ and $\frac{\pi }{2}$
• D. $\frac{\pi }{2}$ and $2\pi$

Answer 1

The correct option is D $\frac{\pi }{2}$ and $2\pi$

$a+\frac{\pi }{2}<2({\tan }^{-1}x+{\cot }^{-1}x)+{\cot }^{-1}x<b$
$a+\frac{\pi }{2}<\pi +{\cot }^{-1}x<b$
$a-\frac{\pi }{2}<{\cot }^{-1}x<b-\pi$
$[\because 0<co{t}^{-1}x<\pi ]$
$a=\frac{\pi }{2}$ and $b=2\pi$