def _futrig(e, **kwargs):
"""Helper for futrig."""
from sympy.simplify.fu import (TR1, TR2, TR3, TR2i, TR10, L, TR10i, TR8,
TR6, TR15, TR16, TR111, TR5, TRmorrie, TR11,
TR14, TR22, TR12)
from sympy.core.compatibility import _nodes
if not e.has(TrigonometricFunction):
return e
if e.is_Mul:
coeff, e = e.as_independent(TrigonometricFunction)
else:
coeff = S.One
Lops = lambda x: (L(x), x.count_ops(), _nodes(x), len(x.args), x.is_Add)
trigs = lambda x: x.has(TrigonometricFunction)
tree = [
identity,
(
TR3, # canonical angles
TR1, # sec-csc -> cos-sin
TR12, # expand tan of sum
lambda x: _eapply(factor, x, trigs),
TR2, # tan-cot -> sin-cos
[identity, lambda x: _eapply(_mexpand, x, trigs)],
TR2i, # sin-cos ratio -> tan
lambda x: _eapply(lambda i: factor(i.normal()), x, trigs),
TR14, # factored identities
TR5, # sin-pow -> cos_pow
TR10, # sin-cos of sums -> sin-cos prod
TR11,
TR6, # reduce double angles and rewrite cos pows
lambda x: _eapply(factor, x, trigs),
TR14, # factored powers of identities
[identity, lambda x: _eapply(_mexpand, x, trigs)],
TRmorrie,
TR10i, # sin-cos products > sin-cos of sums
[identity, TR8], # sin-cos products -> sin-cos of sums
[identity, lambda x: TR2i(TR2(x))], # tan -> sin-cos -> tan
[
lambda x: _eapply(expand_mul, TR5(x), trigs),
lambda x: _eapply(expand_mul, TR15(x), trigs)
], # pos/neg powers of sin
[
lambda x: _eapply(expand_mul, TR6(x), trigs),
lambda x: _eapply(expand_mul, TR16(x), trigs)
], # pos/neg powers of cos
TR111, # tan, sin, cos to neg power -> cot, csc, sec
[identity, TR2i], # sin-cos ratio to tan
[identity, lambda x: _eapply(expand_mul, TR22(x), trigs)
], # tan-cot to sec-csc
TR1,
TR2,
TR2i,
[identity, lambda x: _eapply(factor_terms, TR12(x), trigs)
], # expand tan of sum
)
]
e = greedy(tree, objective=Lops)(e)
return coeff * e
def test_TR5():
assert TR5(sin(x)**2) == -cos(x)**2 + 1
assert TR5(sin(x)**-2) == sin(x)**(-2)
assert TR5(sin(x)**4) == (-cos(x)**2 + 1)**2
