Question

Find cot theta if csc theta= the square root of 5/2 and tan theta>0

Answer 1

Since $\cot^2\theta+1=\csc^2\theta$, you have

$\cot^2\theta=\left(\dfrac{\sqrt5}2\right)^2-1=\dfrac14$
$\cot\theta=\pm\sqrt{\dfrac14}=\pm\dfrac12$

Since $\tan\theta>0$, and $\cot\theta=\dfrac1{\tan\theta}$, you know that $\cot\theta$ is also positive, which means

$\cot\theta=\dfrac12$