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Hm... Its very strange for me but answer is following:

res = -1/2*sin(1/16*pi)*... # expression in your question
import sympy
sympy.simplify(res) # you will get 0

The strange moment is that next code won't get you 0:

res = -1/2*sin(1/16*pi)*... # expression in your question
res.simplify_full().n() # you will get 6.52256026967279e-16 in Sagemathcloud

I check all code in Sagemathcloud. In my opinion the difference is because 2nd way Maxima is used as backend. But now I don't know how to check when you should use Maxima via Sage functions and when - Sympy.

click to hide/show revision 2
check --> checked (past simple)

Hm... Its very strange for me but answer is following:

res = -1/2*sin(1/16*pi)*... # expression in your question
import sympy
sympy.simplify(res) # you will get 0

The strange moment is that next code won't get you 0:

res = -1/2*sin(1/16*pi)*... # expression in your question
res.simplify_full().n() # you will get 6.52256026967279e-16 in Sagemathcloud

I check checked all code in Sagemathcloud. In my opinion the difference is because 2nd way Maxima is used as backend. But now I don't know how to check when you should use Maxima via Sage functions and when - Sympy.

click to hide/show revision 3
2nd way - in 2nd way

Hm... Its very strange for me but answer is following:

res = -1/2*sin(1/16*pi)*... # expression in your question
import sympy
sympy.simplify(res) # you will get 0

The strange moment is that next code won't get you 0:

res = -1/2*sin(1/16*pi)*... # expression in your question
res.simplify_full().n() # you will get 6.52256026967279e-16 in Sagemathcloud

I checked all code in Sagemathcloud. In my opinion the difference is because in 2nd way Maxima is used as backend. But now I don't know how to check when you should use Maxima via Sage functions and when - Sympy.