(B)
Step-by-step explanation:
We know that
[tex]\cot{\theta} = \dfrac{1}{\tan{\theta}} = -\sqrt{3}[/tex]
which means that
[tex]\tan{\theta} = -\dfrac{\sqrt{3}}{3} \Rightarrow \tan^2{\theta} = \dfrac{1}{3}[/tex]
Also, we know that
[tex]\csc{\theta} = \dfrac{1}{\sin{\theta}} = 2[/tex]
which gives us
[tex]\sin{\theta} = \dfrac{1}{2}[/tex]
Using the identity [tex]\cos^2{\theta} + \sin^2{\theta}=1,[/tex] we get
[tex]\cos^2{\theta} = 1- \sin^2{\theta} = 1 - (\frac{1}{2})^2 = \frac{3}{4}[/tex]
We also know that
[tex]\sec^2{\theta} = \dfrac{1}{\cos^2{\theta}} = \dfrac{4}{3}[/tex]
Therefore,
[tex]\sec^2{\theta} + \tan^2{\theta} = \dfrac{4}{3} + \dfrac{1}{3}= \dfrac{5}{3}[/tex]
