Answer
$$\cos(s+t)=-\frac{36}{85}$$
$$\cos(s-t)=\frac{84}{85}$$
Work Step by Step
To find $\cos(s+t)$ and $\cos(s-t)$, cosine sum and difference identities need to be applied.
Yet, both cosine sum and difference identities rely on the fact that you already know $\cos s$, $\sin s$, $\cos t$ and $\sin t$.
Therefore, the first job in this type of exercise must be to find out all 4 values: $\cos s$, $\sin s$, $\cos t$ and $\sin t$.
1) Find $\cos s$, $\sin s$, $\cos t$ and $\sin t$.
As $\cos s$ and $\cos t$ are known, we would use Pythagorean Identities for $\sin$ and $\cos$ to find the sine ones:
$$\sin^2 s=1-\cos^2 s=1-\Big(-\frac{8}{17}\Big)^2=1-\frac{64}{289}=\frac{225}{289}$$
$$\sin s=\pm\frac{15}{17}$$
$$\sin^2 t=1-\cos^2 t=1-\Big(-\frac{3}{5}\Big)^2=1-\frac{9}{25}=\frac{16}{25}$$
$$\sin t=\pm\frac{4}{5}$$
Now about the signs of $\cos s$ and $\cos t$:
As both $s$ and $t$ are in quadrant III, $\sin s\lt0$ and $\sin t\lt0$, so $$\sin s=-\frac{15}{17}\hspace{2cm}\sin t=-\frac{4}{5}$$
2) Now we apply cosine sum and difference identities:
$$\cos(s+t)=\cos s\cos t-\sin s\sin t=\Big(-\frac{8}{17}\Big)\Big(-\frac{3}{5}\Big)-\Big(-\frac{15}{17}\Big)\Big(-\frac{4}{5}\Big)$$
$$\cos(s+t)=\frac{24}{85}-\frac{60}{85}=-\frac{36}{85}$$
$$\cos(s-t)=\cos s\cos t+\sin s\sin t=\Big(-\frac{8}{17}\Big)\Big(-\frac{3}{5}\Big)+\Big(-\frac{15}{17}\Big)\Big(-\frac{4}{5}\Big)$$
$$\cos(s-t)=\frac{24}{85}+\frac{60}{85}=\frac{84}{85}$$
