Unless we are talking about a moment when the displacement is maximum and therefore the cosine term is equal to 1, I don't understand how these two equations can be the same
I understood your explanation for the second question, but my physics textbook shows the equation for acceleration with a cosine term (image attached), so I still don't understand why it doesn't appear in the equation used to solve the first question.
Perhaps that's a very dumb question, but I'm having a hard time to understand why it's possible to omit parts of the equations in solving various problems involving oscillations. Here, for example, the complete equation for acceleration is not used (the part with cosine doesn't appear) and here...