Different looking results upon what seems like a very slight change only happen quite often, specially when dealing with sine/cosine or logarithms.
But C6 cannot simplify to 1 because it isn't.
Your first solve results in what looks like a fraction with equal nominator and denominator, but they differ for one sign - see attached picture.
Furthermore I had assigned dummy values for all variables and had C6 numerically evaluated - far away from 1.
In your manual simplification you obviously overlooked the minus sign in the very first summand of your "Intermediate Equ 1". So you would need a different addition theorem and lastly you would have to change "sin(Phi)*cos(Phi-omega*t)" to "cos(Phi)*sin(Phi-omega*t)" in the next equation.
