trig pythagorean identity

[zlee-personal]

Would it be possible to add a rule that cos^2(x) + sin^2(x) = 1?

[soegaard]

Yes. Suppose you have a polynomial expression P(X,Y) in X=cos(x) and Y=sin(x) then use polynomial-quotient-remainder to calculate P(X,Y) divided by X^2+Y^2. This gives:

P(X,Y) = Q(X,Y) (X^2+Y^2) + R(X,Y).

Now since X^2+Y^2 = 1 we have: P(X,Y) = Q(X,Y) + R(X,Y). So the reduced expression is Q(cos(x),sin(x)) + R(cos(x),sin(x))

I won't have time do this soon - but patches are welcome. FWIW I notice that the functionality is called trigsimp in Maxima.

[BowenFu]

Looks like it is not easy for the polynomial-quotient-remainder approach to handle all the cases.
we can have cos(x), but also cos(pi*x) cos(sqrt(x), so we cannot know what X and Y is in advance.

Instead, I think we can replace cos^2(...) with 1-sin^2(x) before trig-simplify, and recover it after trig-simplify.

[soegaard]

The function (variables u) can return a list of expressions in which u is a variable.
I am thinking that a similar function could determine if an expression u has the form P(X,Y) where X=cos(v) and Y=sin(v).