Let θ be an angle such that cot θ= 12/5 and secθ<0.
Find the exact values of cosθ and cscθ.

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  cos(θ) = -12/13

  csc(θ) = -13/5

Step-by-step explanation:

The relation between the cot and csc is ...

  cot² + 1 = csc²

The signs of the cot and sec tell you this is a 3rd quadrant angle, where cos and csc are both negative.

Then ...

  csc(θ) = -√(1 +(12/5)²) = -√(169/25)

  csc(θ) = -13/5

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  cos(θ) = cot(θ)/csc(θ) = (12/5)/(-13/5)

  cos(θ) = -12/13