Answer:
Option C.
Step-by-step explanation:
It is given that
[tex]\sec \theta=-\dfrac{5}{4}[/tex]
It is also given that [tex]\theta[/tex] is in second quadrant.
Only sin and coses are positive in second quadrant.
We know that
[tex]\tan^2\theta=\sec^2\theta -1[/tex]
[tex]\tan\theta=-\sqrt{\sec^2\theta -1}[/tex] [[tex]\theta[/tex] is in second quadrant]
Substitute [tex]\sec \theta=-\dfrac{5}{4}[/tex] in the above equation.
[tex]\tan\theta=-\sqrt{(-\dfrac{5}{4})^2 -1}[/tex]
[tex]\tan\theta=-\sqrt{\dfrac{25}{16}-1}[/tex]
[tex]\tan\theta=-\sqrt{\dfrac{25-16}{16}}[/tex]
[tex]\tan\theta=-\sqrt{\dfrac{9}{16}}[/tex]
[tex]\tan\theta=-\sqrt{\dfrac{3}{4}}[/tex] ...(1)
Now, we know that
[tex]\cot \theta=\dfrac{1}{\tan \theta}[/tex]
Using (1), we get
[tex]\cot \theta=\dfrac{1}{-\frac{3}{4}}[/tex]
[tex]\cot \theta=-\dfrac{4}{3}[/tex]
Therefore, the correct option is C.
