If α, β are acute angles such that Sin = 1/12 and Cos β = 4/5

Calculate the six trigonometric functions for the sum and differences of the two angles….

Did you mean \sin \alpha = 1/12? There's no angle in the first one, and I'm going to assume you meant \sin\alpha.

Do you have trig identities for the sum and difference of angles? e.g. \cos\left(\alpha + \beta\right) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta) and such?

I suggest that you think of the givens in terms of right triangles, and use SOHCAHTOA and the pythagorean theorem to find \cos\alpha and \sin\beta.

Then use the identities to compute the sine and cosine of the summed and differenced angles, and from there try to get all six trigonometric functions.