Given:
[tex]\begin{gathered} csc\theta=\frac{7}{6} \\ \\ \cos \theta<0 \end{gathered}[/tex]Let's use a trig identity to find the value of tanθ.
Apply the definition of cosecant to find known sides of the unit circle.
We have:
[tex]\csc \theta=\frac{hypotenuse}{opposite}=\frac{7}{6}[/tex]Here, the hypotenuse and opposite sides are known.
Also, find the adjacent side using Pythagorean Theorem:
[tex]\begin{gathered} \text{adjacent}=\sqrt[]{hypotenuse^2-opposite^2} \\ \\ \text{adjacent}=\sqrt[]{7^2-6^2} \\ \\ \text{adjacent}=\sqrt[]{49-36}=\sqrt[]{13} \end{gathered}[/tex]Apply the definition of tangent to find tanθ:
[tex]\begin{gathered} \tan \theta=\frac{opposite}{\text{adjacent}} \\ \\ \text{tan}\theta=\frac{6}{\sqrt[]{13}} \end{gathered}[/tex]Simplify:
[tex]\begin{gathered} \tan \theta=\frac{6}{\sqrt[]{13}}\times\frac{\sqrt[]{13}}{\sqrt[]{13}} \\ \\ \tan \theta=\frac{6\sqrt[]{13}}{13}=1.664 \end{gathered}[/tex]ANSWER:
[tex]\tan \theta=1.664[/tex]