step 1
Find the value of sin
we have that
[tex]\sin ^2(\theta)+\cos ^2(\theta)=1^{}[/tex]substitute the value of cosine
[tex]\begin{gathered} \sin ^2(\theta)+(-\frac{12}{13})^2=1^{} \\ \\ \sin ^2(\theta)^{}=1^{}-(\frac{144}{169}) \\ \\ \sin ^2(\theta)^{}=(\frac{25}{169}) \\ \\ \sin ^{}(\theta)^{}=\frac{5}{13} \end{gathered}[/tex]step 2
Find tan
[tex]\tan (\theta)=\frac{\sin (\theta)}{\cos (\theta)}[/tex]substitute the given values
[tex]\tan (\theta)=-\frac{5}{12}[/tex]step 3
Find cot
[tex]\cot (\theta)=\frac{1}{\tan (\theta)}[/tex]substitute
[tex]\cot (\theta)=-\frac{12}{5}[/tex]step 4
Find sec
[tex]\sec (\theta)=\frac{1}{\cos (\theta)}[/tex]substitute
[tex]\sec (\theta)=-\frac{13}{12}[/tex]step 5
Find csc
[tex]\csc (\theta)=\frac{1}{\sin (\theta)}[/tex]substitute
[tex]\csc (\theta)=\frac{13}{5}[/tex]