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whenever I tried to check this I got "False" for ln[2] and I do not know how to overcome this situation. Actually these two expressions are the elements of a matrix and I can not check either both are same or not

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=== checks for structural sameness, not mathematical equivalence.

You could force both sides to have same structure as follows

Sin[t a2]*(I Cos[t a1] - w Sin[t a1]/a1) === Sin[t a2]*(I Cos[t a1] - w Sin[t a1]/a1)

(* True *)

 Expand[Sin[t a2]*(I Cos[t a1] - w Sin[t a1]/a1)] === 
            Expand[-Sin[t a2]* (-I Cos[t a1] + w Sin[t a1]/a1)]

(* True *)

Mathematica graphics

Another way is simplify lhs - rhs and see if you get zero

Clear["Global`*"];
eq = Sin[t a2]*(I Cos[t a1] - w Sin[t a1]/a1) == -Sin[t a2]* (-I Cos[t a1] + w Sin[t a1]/a1);
lhs = eq /. (lhs_) == (rhs_) :> lhs
rhs = eq /. (lhs_) == (rhs_) :> rhs
Simplify[lhs - rhs]

(* 0 *)

Mathematica graphics

Another way is

Simplify[eq]
(*True*)
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