ANSWER:
[tex]x=141.34,33.69[/tex]
STEP-BY-STEP EXPLANATION:
We have the following equation:
[tex]-8\csc ^2(x)+2\cot \mleft(x\mright)=-23[/tex]
We sovle for x:
[tex]\begin{gathered} \csc ^2(x)=1+\cot ^2(x) \\ \text{ We replacing} \\ -8\cdot(1+\cot ^2(x))+2\cot (x)=-23 \\ -8-8\cot ^2(x)+2\cot (x)+23=0 \\ -8\cot ^2(x)+2\cot (x)+15=0 \\ \text{ Using the substitution method:} \\ \cot (x)=u \\ -8u^2+2u+15=0 \\ -1\cdot(8u^2-2u-15)=0 \\ 8u^2-2u-15=0 \\ -2u=+10u-12u \\ 8u^2+10u-12u-15=0 \\ 2u(4u+5)-3u(4u+5)=0 \\ (4u+5)(2u-3)=0 \\ 4u+5=0\rightarrow u=-\frac{5}{4} \\ 2u+-3=0\rightarrow u=\frac{3}{2} \\ \text{Therefore:} \\ \cot (x)=-\frac{5}{4}\rightarrow x=\cot ^{-1}\mleft(-\frac{5}{4}\mright)\rightarrow x=141.34\degree \\ \cot (x)=\frac{3}{2}\rightarrow x=\cot ^{-1}\mleft(\frac{3}{2}\mright)\rightarrow x=33.69\degree \end{gathered}[/tex]
