[tex]\tan\theta=\dfrac34>0[/tex]
[tex]\cot\theta=\dfrac1{\tan\theta}=\dfrac43>0[/tex]
[tex]\csc^2\theta=1+\cot^2\theta[/tex]
[tex]\csc\theta=\pm\sqrt{1+\left(\dfrac43\right)^2}=\pm\dfrac53[/tex]
Without any more information, it's impossible to tell whether the cosecant is positive or negative. For example, if [tex]\theta=\tan^{-1}\dfrac34[/tex], then
[tex]\tan\left(\tan^{-1}\dfrac34\right)=\dfrac34[/tex]
[tex]\tan\left(\pi+\tan^{-1}\dfrac34\right)=\dfrac34[/tex]
but
[tex]\csc\left(\tan^{-1}\dfrac34\right)=\dfrac53[/tex]
[tex]\csc\left(\pi+\tan^{-1}\dfrac34\right)=-\dfrac53[/tex]
