Question

If the angle $\alpha$ lies in the first quadrant and $\tan \alpha +\cot \alpha =2,$ then the value of $\sqrt{\tan \alpha }+\sqrt{\cot \alpha }$ is

• A. $-4$
• B. $-2$
• C. $2$
• D. $4$

Answer 1

The correct option is C $2$

$(\sqrt{\tan \alpha }+\sqrt{\cot \alpha }{)}^2$

$=\tan \alpha +\cot \alpha +2\left(\sqrt{\tan \alpha \cot \alpha }\right)$

$=2+2=4$ ...(given $\tan \alpha +\cot \alpha =2$)

Taking root on both the sides, we get

$\sqrt{\tan \alpha }+\sqrt{\cot \alpha }=\pm 2$

But $0<\alpha <\frac{\pi }{2}$ and $\sqrt{\tan \alpha }+\sqrt{\cot \alpha }>0$ (square root)

Therefore, $\sqrt{\tan \alpha }+\sqrt{\cot \alpha }=2$

Hence, the answer is $C$.