Respuesta :

(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]

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