It is known that the Hadamard gate is the equivalent of doing a 180 degree rotation about the x + z axis.
I am therefore trying to prove that applying the gate on the state
cos pi/8 |0> + sin pi/8 |1>, which lies on the x + z axis unchanged.
So, on applying the Hadamard on the above, I get
1/root(2)((cos pi/8 + sin pi/8)|0> + (cos pi/8 - sin pi/8)|1>)
So, I did try to take the common global phase out, but I'm unable to prove that this is the same state that I started out with. Any help on how to approach this will be appreciated.