A question related to uniqueness principle theorem.

As you say correctly, the function $z \mapsto |\sin z|^2 + |\cos z|^2$ is not analytic, so the comparision principle cannot be used. Moreover, want you want to prove is wrong, as - for example $$ \def\abs#1{\left|#1\right|} \abs{\sin i}^2 + \abs{\cos i}^2 = \frac{1+e^4}{2e^2} \ne 1. $$