521 lines
20 KiB
Python
521 lines
20 KiB
Python
from itertools import product
|
|
from sympy.core.function import (Subs, count_ops, diff, expand)
|
|
from sympy.core.numbers import (E, I, Rational, pi)
|
|
from sympy.core.singleton import S
|
|
from sympy.core.symbol import (Symbol, symbols)
|
|
from sympy.functions.elementary.exponential import (exp, log)
|
|
from sympy.functions.elementary.hyperbolic import (cosh, coth, sinh, tanh)
|
|
from sympy.functions.elementary.miscellaneous import sqrt
|
|
from sympy.functions.elementary.piecewise import Piecewise
|
|
from sympy.functions.elementary.trigonometric import (cos, cot, sin, tan)
|
|
from sympy.functions.elementary.trigonometric import (acos, asin, atan2)
|
|
from sympy.functions.elementary.trigonometric import (asec, acsc)
|
|
from sympy.functions.elementary.trigonometric import (acot, atan)
|
|
from sympy.integrals.integrals import integrate
|
|
from sympy.matrices.dense import Matrix
|
|
from sympy.simplify.simplify import simplify
|
|
from sympy.simplify.trigsimp import (exptrigsimp, trigsimp)
|
|
|
|
from sympy.testing.pytest import XFAIL
|
|
|
|
from sympy.abc import x, y
|
|
|
|
|
|
|
|
def test_trigsimp1():
|
|
x, y = symbols('x,y')
|
|
|
|
assert trigsimp(1 - sin(x)**2) == cos(x)**2
|
|
assert trigsimp(1 - cos(x)**2) == sin(x)**2
|
|
assert trigsimp(sin(x)**2 + cos(x)**2) == 1
|
|
assert trigsimp(1 + tan(x)**2) == 1/cos(x)**2
|
|
assert trigsimp(1/cos(x)**2 - 1) == tan(x)**2
|
|
assert trigsimp(1/cos(x)**2 - tan(x)**2) == 1
|
|
assert trigsimp(1 + cot(x)**2) == 1/sin(x)**2
|
|
assert trigsimp(1/sin(x)**2 - 1) == 1/tan(x)**2
|
|
assert trigsimp(1/sin(x)**2 - cot(x)**2) == 1
|
|
|
|
assert trigsimp(5*cos(x)**2 + 5*sin(x)**2) == 5
|
|
assert trigsimp(5*cos(x/2)**2 + 2*sin(x/2)**2) == 3*cos(x)/2 + Rational(7, 2)
|
|
|
|
assert trigsimp(sin(x)/cos(x)) == tan(x)
|
|
assert trigsimp(2*tan(x)*cos(x)) == 2*sin(x)
|
|
assert trigsimp(cot(x)**3*sin(x)**3) == cos(x)**3
|
|
assert trigsimp(y*tan(x)**2/sin(x)**2) == y/cos(x)**2
|
|
assert trigsimp(cot(x)/cos(x)) == 1/sin(x)
|
|
|
|
assert trigsimp(sin(x + y) + sin(x - y)) == 2*sin(x)*cos(y)
|
|
assert trigsimp(sin(x + y) - sin(x - y)) == 2*sin(y)*cos(x)
|
|
assert trigsimp(cos(x + y) + cos(x - y)) == 2*cos(x)*cos(y)
|
|
assert trigsimp(cos(x + y) - cos(x - y)) == -2*sin(x)*sin(y)
|
|
assert trigsimp(tan(x + y) - tan(x)/(1 - tan(x)*tan(y))) == \
|
|
sin(y)/(-sin(y)*tan(x) + cos(y)) # -tan(y)/(tan(x)*tan(y) - 1)
|
|
|
|
assert trigsimp(sinh(x + y) + sinh(x - y)) == 2*sinh(x)*cosh(y)
|
|
assert trigsimp(sinh(x + y) - sinh(x - y)) == 2*sinh(y)*cosh(x)
|
|
assert trigsimp(cosh(x + y) + cosh(x - y)) == 2*cosh(x)*cosh(y)
|
|
assert trigsimp(cosh(x + y) - cosh(x - y)) == 2*sinh(x)*sinh(y)
|
|
assert trigsimp(tanh(x + y) - tanh(x)/(1 + tanh(x)*tanh(y))) == \
|
|
sinh(y)/(sinh(y)*tanh(x) + cosh(y))
|
|
|
|
assert trigsimp(cos(0.12345)**2 + sin(0.12345)**2) == 1.0
|
|
e = 2*sin(x)**2 + 2*cos(x)**2
|
|
assert trigsimp(log(e)) == log(2)
|
|
|
|
|
|
def test_trigsimp1a():
|
|
assert trigsimp(sin(2)**2*cos(3)*exp(2)/cos(2)**2) == tan(2)**2*cos(3)*exp(2)
|
|
assert trigsimp(tan(2)**2*cos(3)*exp(2)*cos(2)**2) == sin(2)**2*cos(3)*exp(2)
|
|
assert trigsimp(cot(2)*cos(3)*exp(2)*sin(2)) == cos(3)*exp(2)*cos(2)
|
|
assert trigsimp(tan(2)*cos(3)*exp(2)/sin(2)) == cos(3)*exp(2)/cos(2)
|
|
assert trigsimp(cot(2)*cos(3)*exp(2)/cos(2)) == cos(3)*exp(2)/sin(2)
|
|
assert trigsimp(cot(2)*cos(3)*exp(2)*tan(2)) == cos(3)*exp(2)
|
|
assert trigsimp(sinh(2)*cos(3)*exp(2)/cosh(2)) == tanh(2)*cos(3)*exp(2)
|
|
assert trigsimp(tanh(2)*cos(3)*exp(2)*cosh(2)) == sinh(2)*cos(3)*exp(2)
|
|
assert trigsimp(coth(2)*cos(3)*exp(2)*sinh(2)) == cosh(2)*cos(3)*exp(2)
|
|
assert trigsimp(tanh(2)*cos(3)*exp(2)/sinh(2)) == cos(3)*exp(2)/cosh(2)
|
|
assert trigsimp(coth(2)*cos(3)*exp(2)/cosh(2)) == cos(3)*exp(2)/sinh(2)
|
|
assert trigsimp(coth(2)*cos(3)*exp(2)*tanh(2)) == cos(3)*exp(2)
|
|
|
|
|
|
def test_trigsimp2():
|
|
x, y = symbols('x,y')
|
|
assert trigsimp(cos(x)**2*sin(y)**2 + cos(x)**2*cos(y)**2 + sin(x)**2,
|
|
recursive=True) == 1
|
|
assert trigsimp(sin(x)**2*sin(y)**2 + sin(x)**2*cos(y)**2 + cos(x)**2,
|
|
recursive=True) == 1
|
|
assert trigsimp(
|
|
Subs(x, x, sin(y)**2 + cos(y)**2)) == Subs(x, x, 1)
|
|
|
|
|
|
def test_issue_4373():
|
|
x = Symbol("x")
|
|
assert abs(trigsimp(2.0*sin(x)**2 + 2.0*cos(x)**2) - 2.0) < 1e-10
|
|
|
|
|
|
def test_trigsimp3():
|
|
x, y = symbols('x,y')
|
|
assert trigsimp(sin(x)/cos(x)) == tan(x)
|
|
assert trigsimp(sin(x)**2/cos(x)**2) == tan(x)**2
|
|
assert trigsimp(sin(x)**3/cos(x)**3) == tan(x)**3
|
|
assert trigsimp(sin(x)**10/cos(x)**10) == tan(x)**10
|
|
|
|
assert trigsimp(cos(x)/sin(x)) == 1/tan(x)
|
|
assert trigsimp(cos(x)**2/sin(x)**2) == 1/tan(x)**2
|
|
assert trigsimp(cos(x)**10/sin(x)**10) == 1/tan(x)**10
|
|
|
|
assert trigsimp(tan(x)) == trigsimp(sin(x)/cos(x))
|
|
|
|
|
|
def test_issue_4661():
|
|
a, x, y = symbols('a x y')
|
|
eq = -4*sin(x)**4 + 4*cos(x)**4 - 8*cos(x)**2
|
|
assert trigsimp(eq) == -4
|
|
n = sin(x)**6 + 4*sin(x)**4*cos(x)**2 + 5*sin(x)**2*cos(x)**4 + 2*cos(x)**6
|
|
d = -sin(x)**2 - 2*cos(x)**2
|
|
assert simplify(n/d) == -1
|
|
assert trigsimp(-2*cos(x)**2 + cos(x)**4 - sin(x)**4) == -1
|
|
eq = (- sin(x)**3/4)*cos(x) + (cos(x)**3/4)*sin(x) - sin(2*x)*cos(2*x)/8
|
|
assert trigsimp(eq) == 0
|
|
|
|
|
|
def test_issue_4494():
|
|
a, b = symbols('a b')
|
|
eq = sin(a)**2*sin(b)**2 + cos(a)**2*cos(b)**2*tan(a)**2 + cos(a)**2
|
|
assert trigsimp(eq) == 1
|
|
|
|
|
|
def test_issue_5948():
|
|
a, x, y = symbols('a x y')
|
|
assert trigsimp(diff(integrate(cos(x)/sin(x)**7, x), x)) == \
|
|
cos(x)/sin(x)**7
|
|
|
|
|
|
def test_issue_4775():
|
|
a, x, y = symbols('a x y')
|
|
assert trigsimp(sin(x)*cos(y)+cos(x)*sin(y)) == sin(x + y)
|
|
assert trigsimp(sin(x)*cos(y)+cos(x)*sin(y)+3) == sin(x + y) + 3
|
|
|
|
|
|
def test_issue_4280():
|
|
a, x, y = symbols('a x y')
|
|
assert trigsimp(cos(x)**2 + cos(y)**2*sin(x)**2 + sin(y)**2*sin(x)**2) == 1
|
|
assert trigsimp(a**2*sin(x)**2 + a**2*cos(y)**2*cos(x)**2 + a**2*cos(x)**2*sin(y)**2) == a**2
|
|
assert trigsimp(a**2*cos(y)**2*sin(x)**2 + a**2*sin(y)**2*sin(x)**2) == a**2*sin(x)**2
|
|
|
|
|
|
def test_issue_3210():
|
|
eqs = (sin(2)*cos(3) + sin(3)*cos(2),
|
|
-sin(2)*sin(3) + cos(2)*cos(3),
|
|
sin(2)*cos(3) - sin(3)*cos(2),
|
|
sin(2)*sin(3) + cos(2)*cos(3),
|
|
sin(2)*sin(3) + cos(2)*cos(3) + cos(2),
|
|
sinh(2)*cosh(3) + sinh(3)*cosh(2),
|
|
sinh(2)*sinh(3) + cosh(2)*cosh(3),
|
|
)
|
|
assert [trigsimp(e) for e in eqs] == [
|
|
sin(5),
|
|
cos(5),
|
|
-sin(1),
|
|
cos(1),
|
|
cos(1) + cos(2),
|
|
sinh(5),
|
|
cosh(5),
|
|
]
|
|
|
|
|
|
def test_trigsimp_issues():
|
|
a, x, y = symbols('a x y')
|
|
|
|
# issue 4625 - factor_terms works, too
|
|
assert trigsimp(sin(x)**3 + cos(x)**2*sin(x)) == sin(x)
|
|
|
|
# issue 5948
|
|
assert trigsimp(diff(integrate(cos(x)/sin(x)**3, x), x)) == \
|
|
cos(x)/sin(x)**3
|
|
assert trigsimp(diff(integrate(sin(x)/cos(x)**3, x), x)) == \
|
|
sin(x)/cos(x)**3
|
|
|
|
# check integer exponents
|
|
e = sin(x)**y/cos(x)**y
|
|
assert trigsimp(e) == e
|
|
assert trigsimp(e.subs(y, 2)) == tan(x)**2
|
|
assert trigsimp(e.subs(x, 1)) == tan(1)**y
|
|
|
|
# check for multiple patterns
|
|
assert (cos(x)**2/sin(x)**2*cos(y)**2/sin(y)**2).trigsimp() == \
|
|
1/tan(x)**2/tan(y)**2
|
|
assert trigsimp(cos(x)/sin(x)*cos(x+y)/sin(x+y)) == \
|
|
1/(tan(x)*tan(x + y))
|
|
|
|
eq = cos(2)*(cos(3) + 1)**2/(cos(3) - 1)**2
|
|
assert trigsimp(eq) == eq.factor() # factor makes denom (-1 + cos(3))**2
|
|
assert trigsimp(cos(2)*(cos(3) + 1)**2*(cos(3) - 1)**2) == \
|
|
cos(2)*sin(3)**4
|
|
|
|
# issue 6789; this generates an expression that formerly caused
|
|
# trigsimp to hang
|
|
assert cot(x).equals(tan(x)) is False
|
|
|
|
# nan or the unchanged expression is ok, but not sin(1)
|
|
z = cos(x)**2 + sin(x)**2 - 1
|
|
z1 = tan(x)**2 - 1/cot(x)**2
|
|
n = (1 + z1/z)
|
|
assert trigsimp(sin(n)) != sin(1)
|
|
eq = x*(n - 1) - x*n
|
|
assert trigsimp(eq) is S.NaN
|
|
assert trigsimp(eq, recursive=True) is S.NaN
|
|
assert trigsimp(1).is_Integer
|
|
|
|
assert trigsimp(-sin(x)**4 - 2*sin(x)**2*cos(x)**2 - cos(x)**4) == -1
|
|
|
|
|
|
def test_trigsimp_issue_2515():
|
|
x = Symbol('x')
|
|
assert trigsimp(x*cos(x)*tan(x)) == x*sin(x)
|
|
assert trigsimp(-sin(x) + cos(x)*tan(x)) == 0
|
|
|
|
|
|
def test_trigsimp_issue_3826():
|
|
assert trigsimp(tan(2*x).expand(trig=True)) == tan(2*x)
|
|
|
|
|
|
def test_trigsimp_issue_4032():
|
|
n = Symbol('n', integer=True, positive=True)
|
|
assert trigsimp(2**(n/2)*cos(pi*n/4)/2 + 2**(n - 1)/2) == \
|
|
2**(n/2)*cos(pi*n/4)/2 + 2**n/4
|
|
|
|
|
|
def test_trigsimp_issue_7761():
|
|
assert trigsimp(cosh(pi/4)) == cosh(pi/4)
|
|
|
|
|
|
def test_trigsimp_noncommutative():
|
|
x, y = symbols('x,y')
|
|
A, B = symbols('A,B', commutative=False)
|
|
|
|
assert trigsimp(A - A*sin(x)**2) == A*cos(x)**2
|
|
assert trigsimp(A - A*cos(x)**2) == A*sin(x)**2
|
|
assert trigsimp(A*sin(x)**2 + A*cos(x)**2) == A
|
|
assert trigsimp(A + A*tan(x)**2) == A/cos(x)**2
|
|
assert trigsimp(A/cos(x)**2 - A) == A*tan(x)**2
|
|
assert trigsimp(A/cos(x)**2 - A*tan(x)**2) == A
|
|
assert trigsimp(A + A*cot(x)**2) == A/sin(x)**2
|
|
assert trigsimp(A/sin(x)**2 - A) == A/tan(x)**2
|
|
assert trigsimp(A/sin(x)**2 - A*cot(x)**2) == A
|
|
|
|
assert trigsimp(y*A*cos(x)**2 + y*A*sin(x)**2) == y*A
|
|
|
|
assert trigsimp(A*sin(x)/cos(x)) == A*tan(x)
|
|
assert trigsimp(A*tan(x)*cos(x)) == A*sin(x)
|
|
assert trigsimp(A*cot(x)**3*sin(x)**3) == A*cos(x)**3
|
|
assert trigsimp(y*A*tan(x)**2/sin(x)**2) == y*A/cos(x)**2
|
|
assert trigsimp(A*cot(x)/cos(x)) == A/sin(x)
|
|
|
|
assert trigsimp(A*sin(x + y) + A*sin(x - y)) == 2*A*sin(x)*cos(y)
|
|
assert trigsimp(A*sin(x + y) - A*sin(x - y)) == 2*A*sin(y)*cos(x)
|
|
assert trigsimp(A*cos(x + y) + A*cos(x - y)) == 2*A*cos(x)*cos(y)
|
|
assert trigsimp(A*cos(x + y) - A*cos(x - y)) == -2*A*sin(x)*sin(y)
|
|
|
|
assert trigsimp(A*sinh(x + y) + A*sinh(x - y)) == 2*A*sinh(x)*cosh(y)
|
|
assert trigsimp(A*sinh(x + y) - A*sinh(x - y)) == 2*A*sinh(y)*cosh(x)
|
|
assert trigsimp(A*cosh(x + y) + A*cosh(x - y)) == 2*A*cosh(x)*cosh(y)
|
|
assert trigsimp(A*cosh(x + y) - A*cosh(x - y)) == 2*A*sinh(x)*sinh(y)
|
|
|
|
assert trigsimp(A*cos(0.12345)**2 + A*sin(0.12345)**2) == 1.0*A
|
|
|
|
|
|
def test_hyperbolic_simp():
|
|
x, y = symbols('x,y')
|
|
|
|
assert trigsimp(sinh(x)**2 + 1) == cosh(x)**2
|
|
assert trigsimp(cosh(x)**2 - 1) == sinh(x)**2
|
|
assert trigsimp(cosh(x)**2 - sinh(x)**2) == 1
|
|
assert trigsimp(1 - tanh(x)**2) == 1/cosh(x)**2
|
|
assert trigsimp(1 - 1/cosh(x)**2) == tanh(x)**2
|
|
assert trigsimp(tanh(x)**2 + 1/cosh(x)**2) == 1
|
|
assert trigsimp(coth(x)**2 - 1) == 1/sinh(x)**2
|
|
assert trigsimp(1/sinh(x)**2 + 1) == 1/tanh(x)**2
|
|
assert trigsimp(coth(x)**2 - 1/sinh(x)**2) == 1
|
|
|
|
assert trigsimp(5*cosh(x)**2 - 5*sinh(x)**2) == 5
|
|
assert trigsimp(5*cosh(x/2)**2 - 2*sinh(x/2)**2) == 3*cosh(x)/2 + Rational(7, 2)
|
|
|
|
assert trigsimp(sinh(x)/cosh(x)) == tanh(x)
|
|
assert trigsimp(tanh(x)) == trigsimp(sinh(x)/cosh(x))
|
|
assert trigsimp(cosh(x)/sinh(x)) == 1/tanh(x)
|
|
assert trigsimp(2*tanh(x)*cosh(x)) == 2*sinh(x)
|
|
assert trigsimp(coth(x)**3*sinh(x)**3) == cosh(x)**3
|
|
assert trigsimp(y*tanh(x)**2/sinh(x)**2) == y/cosh(x)**2
|
|
assert trigsimp(coth(x)/cosh(x)) == 1/sinh(x)
|
|
|
|
for a in (pi/6*I, pi/4*I, pi/3*I):
|
|
assert trigsimp(sinh(a)*cosh(x) + cosh(a)*sinh(x)) == sinh(x + a)
|
|
assert trigsimp(-sinh(a)*cosh(x) + cosh(a)*sinh(x)) == sinh(x - a)
|
|
|
|
e = 2*cosh(x)**2 - 2*sinh(x)**2
|
|
assert trigsimp(log(e)) == log(2)
|
|
|
|
# issue 19535:
|
|
assert trigsimp(sqrt(cosh(x)**2 - 1)) == sqrt(sinh(x)**2)
|
|
|
|
assert trigsimp(cosh(x)**2*cosh(y)**2 - cosh(x)**2*sinh(y)**2 - sinh(x)**2,
|
|
recursive=True) == 1
|
|
assert trigsimp(sinh(x)**2*sinh(y)**2 - sinh(x)**2*cosh(y)**2 + cosh(x)**2,
|
|
recursive=True) == 1
|
|
|
|
assert abs(trigsimp(2.0*cosh(x)**2 - 2.0*sinh(x)**2) - 2.0) < 1e-10
|
|
|
|
assert trigsimp(sinh(x)**2/cosh(x)**2) == tanh(x)**2
|
|
assert trigsimp(sinh(x)**3/cosh(x)**3) == tanh(x)**3
|
|
assert trigsimp(sinh(x)**10/cosh(x)**10) == tanh(x)**10
|
|
assert trigsimp(cosh(x)**3/sinh(x)**3) == 1/tanh(x)**3
|
|
|
|
assert trigsimp(cosh(x)/sinh(x)) == 1/tanh(x)
|
|
assert trigsimp(cosh(x)**2/sinh(x)**2) == 1/tanh(x)**2
|
|
assert trigsimp(cosh(x)**10/sinh(x)**10) == 1/tanh(x)**10
|
|
|
|
assert trigsimp(x*cosh(x)*tanh(x)) == x*sinh(x)
|
|
assert trigsimp(-sinh(x) + cosh(x)*tanh(x)) == 0
|
|
|
|
assert tan(x) != 1/cot(x) # cot doesn't auto-simplify
|
|
|
|
assert trigsimp(tan(x) - 1/cot(x)) == 0
|
|
assert trigsimp(3*tanh(x)**7 - 2/coth(x)**7) == tanh(x)**7
|
|
|
|
|
|
def test_trigsimp_groebner():
|
|
from sympy.simplify.trigsimp import trigsimp_groebner
|
|
|
|
c = cos(x)
|
|
s = sin(x)
|
|
ex = (4*s*c + 12*s + 5*c**3 + 21*c**2 + 23*c + 15)/(
|
|
-s*c**2 + 2*s*c + 15*s + 7*c**3 + 31*c**2 + 37*c + 21)
|
|
resnum = (5*s - 5*c + 1)
|
|
resdenom = (8*s - 6*c)
|
|
results = [resnum/resdenom, (-resnum)/(-resdenom)]
|
|
assert trigsimp_groebner(ex) in results
|
|
assert trigsimp_groebner(s/c, hints=[tan]) == tan(x)
|
|
assert trigsimp_groebner(c*s) == c*s
|
|
assert trigsimp((-s + 1)/c + c/(-s + 1),
|
|
method='groebner') == 2/c
|
|
assert trigsimp((-s + 1)/c + c/(-s + 1),
|
|
method='groebner', polynomial=True) == 2/c
|
|
|
|
# Test quick=False works
|
|
assert trigsimp_groebner(ex, hints=[2]) in results
|
|
assert trigsimp_groebner(ex, hints=[int(2)]) in results
|
|
|
|
# test "I"
|
|
assert trigsimp_groebner(sin(I*x)/cos(I*x), hints=[tanh]) == I*tanh(x)
|
|
|
|
# test hyperbolic / sums
|
|
assert trigsimp_groebner((tanh(x)+tanh(y))/(1+tanh(x)*tanh(y)),
|
|
hints=[(tanh, x, y)]) == tanh(x + y)
|
|
|
|
|
|
def test_issue_2827_trigsimp_methods():
|
|
measure1 = lambda expr: len(str(expr))
|
|
measure2 = lambda expr: -count_ops(expr)
|
|
# Return the most complicated result
|
|
expr = (x + 1)/(x + sin(x)**2 + cos(x)**2)
|
|
ans = Matrix([1])
|
|
M = Matrix([expr])
|
|
assert trigsimp(M, method='fu', measure=measure1) == ans
|
|
assert trigsimp(M, method='fu', measure=measure2) != ans
|
|
# all methods should work with Basic expressions even if they
|
|
# aren't Expr
|
|
M = Matrix.eye(1)
|
|
assert all(trigsimp(M, method=m) == M for m in
|
|
'fu matching groebner old'.split())
|
|
# watch for E in exptrigsimp, not only exp()
|
|
eq = 1/sqrt(E) + E
|
|
assert exptrigsimp(eq) == eq
|
|
|
|
def test_issue_15129_trigsimp_methods():
|
|
t1 = Matrix([sin(Rational(1, 50)), cos(Rational(1, 50)), 0])
|
|
t2 = Matrix([sin(Rational(1, 25)), cos(Rational(1, 25)), 0])
|
|
t3 = Matrix([cos(Rational(1, 25)), sin(Rational(1, 25)), 0])
|
|
r1 = t1.dot(t2)
|
|
r2 = t1.dot(t3)
|
|
assert trigsimp(r1) == cos(Rational(1, 50))
|
|
assert trigsimp(r2) == sin(Rational(3, 50))
|
|
|
|
def test_exptrigsimp():
|
|
def valid(a, b):
|
|
from sympy.core.random import verify_numerically as tn
|
|
if not (tn(a, b) and a == b):
|
|
return False
|
|
return True
|
|
|
|
assert exptrigsimp(exp(x) + exp(-x)) == 2*cosh(x)
|
|
assert exptrigsimp(exp(x) - exp(-x)) == 2*sinh(x)
|
|
assert exptrigsimp((2*exp(x)-2*exp(-x))/(exp(x)+exp(-x))) == 2*tanh(x)
|
|
assert exptrigsimp((2*exp(2*x)-2)/(exp(2*x)+1)) == 2*tanh(x)
|
|
e = [cos(x) + I*sin(x), cos(x) - I*sin(x),
|
|
cosh(x) - sinh(x), cosh(x) + sinh(x)]
|
|
ok = [exp(I*x), exp(-I*x), exp(-x), exp(x)]
|
|
assert all(valid(i, j) for i, j in zip(
|
|
[exptrigsimp(ei) for ei in e], ok))
|
|
|
|
ue = [cos(x) + sin(x), cos(x) - sin(x),
|
|
cosh(x) + I*sinh(x), cosh(x) - I*sinh(x)]
|
|
assert [exptrigsimp(ei) == ei for ei in ue]
|
|
|
|
res = []
|
|
ok = [y*tanh(1), 1/(y*tanh(1)), I*y*tan(1), -I/(y*tan(1)),
|
|
y*tanh(x), 1/(y*tanh(x)), I*y*tan(x), -I/(y*tan(x)),
|
|
y*tanh(1 + I), 1/(y*tanh(1 + I))]
|
|
for a in (1, I, x, I*x, 1 + I):
|
|
w = exp(a)
|
|
eq = y*(w - 1/w)/(w + 1/w)
|
|
res.append(simplify(eq))
|
|
res.append(simplify(1/eq))
|
|
assert all(valid(i, j) for i, j in zip(res, ok))
|
|
|
|
for a in range(1, 3):
|
|
w = exp(a)
|
|
e = w + 1/w
|
|
s = simplify(e)
|
|
assert s == exptrigsimp(e)
|
|
assert valid(s, 2*cosh(a))
|
|
e = w - 1/w
|
|
s = simplify(e)
|
|
assert s == exptrigsimp(e)
|
|
assert valid(s, 2*sinh(a))
|
|
|
|
def test_exptrigsimp_noncommutative():
|
|
a,b = symbols('a b', commutative=False)
|
|
x = Symbol('x', commutative=True)
|
|
assert exp(a + x) == exptrigsimp(exp(a)*exp(x))
|
|
p = exp(a)*exp(b) - exp(b)*exp(a)
|
|
assert p == exptrigsimp(p) != 0
|
|
|
|
def test_powsimp_on_numbers():
|
|
assert 2**(Rational(1, 3) - 2) == 2**Rational(1, 3)/4
|
|
|
|
|
|
@XFAIL
|
|
def test_issue_6811_fail():
|
|
# from doc/src/modules/physics/mechanics/examples.rst, the current `eq`
|
|
# at Line 576 (in different variables) was formerly the equivalent and
|
|
# shorter expression given below...it would be nice to get the short one
|
|
# back again
|
|
xp, y, x, z = symbols('xp, y, x, z')
|
|
eq = 4*(-19*sin(x)*y + 5*sin(3*x)*y + 15*cos(2*x)*z - 21*z)*xp/(9*cos(x) - 5*cos(3*x))
|
|
assert trigsimp(eq) == -2*(2*cos(x)*tan(x)*y + 3*z)*xp/cos(x)
|
|
|
|
|
|
def test_Piecewise():
|
|
e1 = x*(x + y) - y*(x + y)
|
|
e2 = sin(x)**2 + cos(x)**2
|
|
e3 = expand((x + y)*y/x)
|
|
# s1 = simplify(e1)
|
|
s2 = simplify(e2)
|
|
# s3 = simplify(e3)
|
|
|
|
# trigsimp tries not to touch non-trig containing args
|
|
assert trigsimp(Piecewise((e1, e3 < e2), (e3, True))) == \
|
|
Piecewise((e1, e3 < s2), (e3, True))
|
|
|
|
|
|
def test_issue_21594():
|
|
assert simplify(exp(Rational(1,2)) + exp(Rational(-1,2))) == cosh(S.Half)*2
|
|
|
|
|
|
def test_trigsimp_old():
|
|
x, y = symbols('x,y')
|
|
|
|
assert trigsimp(1 - sin(x)**2, old=True) == cos(x)**2
|
|
assert trigsimp(1 - cos(x)**2, old=True) == sin(x)**2
|
|
assert trigsimp(sin(x)**2 + cos(x)**2, old=True) == 1
|
|
assert trigsimp(1 + tan(x)**2, old=True) == 1/cos(x)**2
|
|
assert trigsimp(1/cos(x)**2 - 1, old=True) == tan(x)**2
|
|
assert trigsimp(1/cos(x)**2 - tan(x)**2, old=True) == 1
|
|
assert trigsimp(1 + cot(x)**2, old=True) == 1/sin(x)**2
|
|
assert trigsimp(1/sin(x)**2 - cot(x)**2, old=True) == 1
|
|
|
|
assert trigsimp(5*cos(x)**2 + 5*sin(x)**2, old=True) == 5
|
|
|
|
assert trigsimp(sin(x)/cos(x), old=True) == tan(x)
|
|
assert trigsimp(2*tan(x)*cos(x), old=True) == 2*sin(x)
|
|
assert trigsimp(cot(x)**3*sin(x)**3, old=True) == cos(x)**3
|
|
assert trigsimp(y*tan(x)**2/sin(x)**2, old=True) == y/cos(x)**2
|
|
assert trigsimp(cot(x)/cos(x), old=True) == 1/sin(x)
|
|
|
|
assert trigsimp(sin(x + y) + sin(x - y), old=True) == 2*sin(x)*cos(y)
|
|
assert trigsimp(sin(x + y) - sin(x - y), old=True) == 2*sin(y)*cos(x)
|
|
assert trigsimp(cos(x + y) + cos(x - y), old=True) == 2*cos(x)*cos(y)
|
|
assert trigsimp(cos(x + y) - cos(x - y), old=True) == -2*sin(x)*sin(y)
|
|
|
|
assert trigsimp(sinh(x + y) + sinh(x - y), old=True) == 2*sinh(x)*cosh(y)
|
|
assert trigsimp(sinh(x + y) - sinh(x - y), old=True) == 2*sinh(y)*cosh(x)
|
|
assert trigsimp(cosh(x + y) + cosh(x - y), old=True) == 2*cosh(x)*cosh(y)
|
|
assert trigsimp(cosh(x + y) - cosh(x - y), old=True) == 2*sinh(x)*sinh(y)
|
|
|
|
assert trigsimp(cos(0.12345)**2 + sin(0.12345)**2, old=True) == 1.0
|
|
|
|
assert trigsimp(sin(x)/cos(x), old=True, method='combined') == tan(x)
|
|
assert trigsimp(sin(x)/cos(x), old=True, method='groebner') == sin(x)/cos(x)
|
|
assert trigsimp(sin(x)/cos(x), old=True, method='groebner', hints=[tan]) == tan(x)
|
|
|
|
assert trigsimp(1-sin(sin(x)**2+cos(x)**2)**2, old=True, deep=True) == cos(1)**2
|
|
|
|
|
|
def test_trigsimp_inverse():
|
|
alpha = symbols('alpha')
|
|
s, c = sin(alpha), cos(alpha)
|
|
|
|
for finv in [asin, acos, asec, acsc, atan, acot]:
|
|
f = finv.inverse(None)
|
|
assert alpha == trigsimp(finv(f(alpha)), inverse=True)
|
|
|
|
# test atan2(cos, sin), atan2(sin, cos), etc...
|
|
for a, b in [[c, s], [s, c]]:
|
|
for i, j in product([-1, 1], repeat=2):
|
|
angle = atan2(i*b, j*a)
|
|
angle_inverted = trigsimp(angle, inverse=True)
|
|
assert angle_inverted != angle # assures simplification happened
|
|
assert sin(angle_inverted) == trigsimp(sin(angle))
|
|
assert cos(angle_inverted) == trigsimp(cos(angle))
|