ANSWER:
[tex]\frac{3\sqrt{13}}{13}[/tex]STEP-BY-STEP EXPLANATION:
We have the following expression:
[tex]\cos\left(\tan^{-1}\left(-\frac{2}{3}\right)\right)[/tex][tex]\begin{gathered} \cos\left(\tan^{-1}\left(-\frac{2}{3}\right)\right)=\cos\left(\arctan\left(-\frac{2}{3}\right)\right) \\ \\ \cos\left(-\arctan\left(\frac{2}{3}\right)\right)=\cos\left(\arctan\left(\frac{2}{3}\right)\right) \\ \\ \text{ we have the following} \\ \\ \cos \left(\arctan \left(x\right)\right)=\frac{\sqrt{1+x^2}}{1+x^2} \\ \\ \text{ we replacing:} \\ \\ \cos\left(\arctan\left(\frac{2}{3}\right)\right)=\frac{\sqrt{1+\left(\frac{2}{3}\right)^2}}{1+\left(\frac{2}{3}\right)^2}=\frac{\sqrt{1+\left(\frac{2}{3}\right)^2}}{1+\frac{2^2}{3^2}}=\frac{\sqrt{1+\frac{4}{9}}}{1+\frac{4}{9}}=\frac{\sqrt{\frac{13}{9}}}{\frac{13}{9}}=\frac{\frac{1}{3}\sqrt{13}}{\frac{13}{9}}=\frac{\sqrt{13}}{3\cdot\frac{13}{9}}=\frac{\sqrt{13}}{\frac{13}{3}}=\frac{3\sqrt{13}}{13} \end{gathered}[/tex]