# Trigonometric equality (with arctan and arcsin)

Hi,

I found a strange issue:

ex = (arctan(1/2) + arcsin(sqrt(5)/5) == arctan(4/3))
bool(ex)


And it gives False.

A simple numeric approximation allows to see that it's true, and wolfram alpha for instance gives the correct answer. But why is it like that and how can I fix it?

Thanks

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In sagecell, giac('simplify(tExpand((arctan(1/2) + arcsin(sqrt(5)/5) - arctan(4/3))))') returns $0$. tExpand is a shortcut for expand trascendental expressions.

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Actually, Sage returns False whenever it does not manage to prove that an equality is true or false. Therefore only True is a mathematically meaningful output for such a test.

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Oh, that's problematical. It's a shame not to be able to verify such an equality. Really, there is no way?

I tried:

sage: tan( arctan(1/2) + arcsin(sqrt(5)/5) - arctan(4/3) ).simplify_full()

0

Usually, it is a good idea to put all expressions on one side. The symbolic expression simplifier has a simpler work.

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1

Your way is also working, however I find weird having to add a 'tan(...)' to the expression to prove that it's equal to 0. Indeed, tan(0)=0 but that does not always work (tan(2k*pi)=0 too)