Question

The value of $4({\cot }^{-1}3+{\text{cosec}}^{-1}\sqrt{5})$ will be

• A. $\frac{\pi }{2}$
• B. $\pi$
• C. $2\pi$
• D. $\frac{\pi }{4}$

Answer 1

The correct option is B $\pi$

Let ${\text{cosec}}^{-1}\sqrt{5}=\theta$ then $\text{cosec}\theta =\sqrt{5}$

$\because {\cot }^2\theta ={\text{cosec}}^2\theta -1=5-1=4$

$\therefore \cot \theta =2.$ Therefore, $\theta ={\cot }^{-1}2$

$\therefore {\text{cosec}}^{-1}\sqrt{5}=\theta ={\cot }^{-1}2$

Therefore, $4({\cot }^{-1}3+{\cot }^{-1}2)$

$=4({\tan }^{-1}\frac{1}{3}+{\tan }^{-1}\frac{1}{2})=4{\tan }^{-1}\left\{\frac{\frac{1}{3}+\frac{1}{2}}{1-\frac{1}{3}.\frac{1}{2}}\right\}$

$=4{\tan }^{-1}\left(\frac{5/6}{5/6}\right)=4{\tan }^{-1}\left(1\right)=4\left(\frac{1}{4}\pi \right)=\pi$