Transfer matrix for a finite length? (Quantum mechanics)

[Schwarzschild90]

Homework Statement

I'm struggling to find a solution to exercise (*b). I have uploaded a pdf of the assignment.

Please advise me at your convenience.

Homework Equations

$$x(x_l^+) = T(x_l^+, x_l^-)x(x_l^-)$$

The Attempt at a Solution

$$x(a^-) = \frac{\psi(a^-)}{\psi(a^-)} , T(a^+, a^-) \left( \frac{\psi(a^-)}{\psi(a^-)} \right)$$

Now, the final result is a matrix of which the rows and columns consist of sine and cosine terms. I have an intuition for why the matrix looks like that, but do not know how to progress to that point. My intution tells me that since the wave function might be restated as a linear combination of vectors, then I could use Euler's identity to reexpress it.

 

[BigJDubs]

However, I'm not sure how to do that, or if this is the right approach. Any help would be appreciated.