simplify works better than trigsimp

[meganly]

simplify does better than trigsimp in simplifying the following expression.

>>> from sympy import trigsimp, simplify, symbols, cos, sin, cot
>>> x = symbols('x')
>>> expr = 1/(sin(x)*cos(x))-cot(x)
>>> simplify(expr)
tan(x)
>>> trigsimp(expr)
-cot(x) + 2/sin(2*x)

As @asmeurer suspected, simplify writes the expression as a combined fraction first

>>> trigsimp(cancel(expr))
tan(x)

It seems that trigsimp should try cancel() first somewhere in here

sympy/sympy/simplify/trigsimp.py

Lines 1132 to 1137 in fed3bb8

	tree = [identity,
	(
	TR3, # canonical angles
	TR1, # sec-csc -> cos-sin
	TR12, # expand tan of sum
	lambda x: _eapply(factor, x, trigs),

[asmeurer]

This would also be a good use case for the groebner method, but it doesn't seem to know about cot.

[meganly]

True, but the default trigsimp algorithm doesn't call the groebner method.

Would adding lambda x: _eapply(cancel, x, trigs), to the tree shown in the code above make trigsimp also try canceling? I'm a little confused how the code to construct the tree works.

[asmeurer]

greedy tries the functions greedily to minimize the objective function which is defined above that code.

I think a problem that you would run into is that the objective is the number of trig functions (L) and then the total number of operations. But cancel actually increases the number of operations

>>> expr = 1/(sin(x)*cos(x))-cot(x)
>>> expr.count_ops()
6
>>> cancel(expr).count_ops()
11

So my guess from reading the code is that it wouldn't work. But I also didn't try it, so I could be wrong.

[asmeurer]

cancel also increases the number of trig functions

>>> from sympy.simplify.fu import L
>>> L(expr)
3
>>> L(cancel(expr))
5