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As for your first question, this is due to the fact that simplification is done by maxima and maxima is not able to deal with unicode symbolic variables. A suggestion is to reserve unicode characters to the Python names and keep ascii strings for symbols:

sage: θ1 = function('theta_1')(t)

Then maxima will be able to handle it correctly, and if you still want to display them with greek letters, you can to

%display latex

so that when you type:

sage: θ1

the output will be $\theta_1(t)$

As fot the second question, you can do:

sage: K.substitute({θ1:z})
cos(z)^2 + sin(z)^2

sage: K = K.substitute({θ1:z})
sage: K
cos(z)^2 + sin(z)^2
sage: K.full_simplify()
1