Merge pull request #417 from JuliaSymbolics/s/double-angle

add double angle formulas

Author: shashi

src/simplify_rules.jl

@@ -57,10 +57,16 @@ let
     ]
 
     TRIG_EXP_RULES = [
+        @acrule(~r*~x::has_trig_exp + ~r*~y => ~r*(~x + ~y))
+        @acrule(~r*~x::has_trig_exp + -1*~r*~y => ~r*(~x - ~y))
         @acrule(sin(~x)^2 + cos(~x)^2 => one(~x))
         @acrule(sin(~x)^2 + -1        => cos(~x)^2)
         @acrule(cos(~x)^2 + -1        => sin(~x)^2)
 
+        @acrule(cos(~x)^2 + -1*sin(~x)^2 => cos(2 * ~x))
+        @acrule(cos(~x)^2 + -1*sin(~x)^2 => cos(2 * ~x))
+        @acrule(cos(~x) * sin(~x) => sin(2 * ~x)/2)
+
         @acrule(tan(~x)^2 + -1*sec(~x)^2 => one(~x))
         @acrule(tan(~x)^2 +  1 => sec(~x)^2)
         @acrule(sec(~x)^2 + -1 => tan(~x)^2)

test/polyform.jl

@@ -11,7 +11,7 @@ include("utils.jl")
                                   (y*(2x - 3y + 3z) +
                                    x*(x + z)))) == repr(simplify_fractions((1 + x + 4z) / (x + 3.0y)))
     @test simplify_fractions(x/(x+3) + 3/(x+3)) == 1
-    @test repr(simplify(simplify_fractions(cos(x)/sin(x) + sin(x)/cos(x)))) == "1 / (cos(x)*sin(x))"
+    @test repr(simplify(simplify_fractions(cos(x)/sin(x) + sin(x)/cos(x)))) == "2 / sin(2x)"
 end
 
 @testset "expand" begin

test/rulesets.jl

@@ -88,6 +88,12 @@ end
     @eqtest simplify(1 + y + cot(x)^2) == csc(x)^2 + y
 end
 
+@testset "Double angle formulas" begin
+    @syms r x
+    @eqtest simplify(r*cos(x/2)^2 - r*sin(x/2)^2) == r*cos(x)
+    @eqtest simplify(2cos(x)*sin(x)) == sin(2x)
+end
+
 @testset "Exponentials" begin
     @syms a::Real b::Real
     @eqtest simplify(exp(a)*exp(b)) == simplify(exp(a+b))