Generally, the problems you are discussing involve working with trig identities. These are part of the core learning in a trigonometry class. They are most useful when manipulating trig expressions into a form which, for some reason, makes the underlying structure of the model more evident. Most often, this is something being done (please forgive me) with pencil and paper. The reason for this is that, while Mathematica is a marvelous tool, all such tools still fall far short of the capabilities of the human mind.

I would guess that one reason for that lies in the words "more evident." In working physics problems, for example, those who have done it for a long time (and some right away) develop an intuition that tells them when a particular rearrangement of an expression is moving toward a more useful form. They know they are trying to get to a form which will make certain characteristics of the model evident. An example might be the derivation of a wave equation from Maxwell's equations. He knew immediately that what he had was astounding.

Thanks for the answer, but how should i know the next time if i have a problem? must i learn the wolframalpha langue that you suggested?

I tested with a new question sin^2(x)- cos^2(x) =1 in the book it says the answer are x = 90°+ n x 360° . How should i think here? Iam really sorry if iam bothering you.

a) I do not know what you mean by "develop" here.

b) sin(x+55°) is simpler than sin x cos(55°) + cos(x) sin(55°).

c) Is the result that Wolfram|Alpha gives you actually wrong, or just different? It gives me cos((7 pi)/36-x), which appears to be equivalent.

In[1]:= FullSimplify[ Cos[(7 Pi)/36 - x] -  Sin [x + 55 \[Degree]],  Element[x, Reals]] 
Out[1]= 0 

Can you show us the output you get, either with a screen shot, or by clicking on it, then clicking the A in the little menu that pops down, then copying the text version?