I wonder if anyone could help me with this exercise?

From the hints I take it that there are three H gates and 3 T gates and that they must alternate. This seems to give two possibilities - THTHTH or HTHTHT. I assume that in each case T may either be itself or its adjoint.

However, I can’t reproduce the unitary matrix. It is the third term in each case with the i factored out that I can’t reproduce.

I believe that I will be able to follow the steps if set out, but my linear algebra is rusty - in particular with multiplying out the terms with e to the power i, so I am unlikely to get there on my own.

Any help would be greatly appreciated.