Answer:
3rd option
Step-by-step explanation:
Using the identities
cot x = [tex]\frac{1}{tanx}[/tex]
csc² x = 1 + cot² x
Given
tanθ = [tex]\sqrt{3}[/tex] , then cotθ = [tex]\frac{1}{\sqrt{3} }[/tex]
csc²θ = 1 + ([tex]\frac{1}{\sqrt{3} }[/tex] )² = 1 + [tex]\frac{1}{3}[/tex] = [tex]\frac{4}{3}[/tex]
cscθ = ± [tex]\sqrt{\frac{4}{3} }[/tex] = ± [tex]\frac{2}{\sqrt{3} }[/tex]
Since θ is in 3rd quadrant, then cscθ < 0
cscθ = - [tex]\frac{2}{\sqrt{3} }[/tex] × [tex]\frac{\sqrt{3} }{\sqrt{3} }[/tex] = - [tex]\frac{2\sqrt{3} }{3}[/tex]