diff --git a/src/simplify_rules.jl b/src/simplify_rules.jl
index 88a1a6482..b12a9f272 100644
--- a/src/simplify_rules.jl
+++ b/src/simplify_rules.jl
@@ -43,7 +43,6 @@ let
 
     MUL_DISTRIBUTE = @ordered_acrule((~x)^(~n) * (~x)^(~m) => (~x)^(~n + ~m))
 
-
     CANONICALIZE_POW = [
         @rule(^(*(~~x), ~y::_isinteger) => *(map(a->pow(a, ~y), ~~x)...))
         @rule((((~x)^(~p::_isinteger))^(~q::_isinteger)) => (~x)^((~p)*(~q)))
@@ -91,6 +90,13 @@ let
         @acrule(cot(~x)^2 +  1 => csc(~x)^2)
         @acrule(csc(~x)^2 + -1 => cot(~x)^2)
 
+        @acrule(cosh(~x)^2 + -1*sinh(~x)^2 => one(~x))
+        @acrule(cosh(~x)^2 + -1            => sinh(~x)^2)
+        @acrule(sinh(~x)^2 +  1            => cosh(~x)^2)
+
+        @acrule(cosh(~x)^2 + sinh(~x)^2 => cosh(2 * ~x))
+        @acrule(cosh(~x) * sinh(~x) => sinh(2 * ~x)/2)
+
         @acrule(exp(~x) * exp(~y) => _iszero(~x + ~y) ? 1 : exp(~x + ~y))
         @rule(exp(~x)^(~y) => exp(~x * ~y))
     ]
diff --git a/src/utils.jl b/src/utils.jl
index 90e0de5cc..acf9e92d6 100644
--- a/src/utils.jl
+++ b/src/utils.jl
@@ -36,10 +36,10 @@ pow(x::Symbolic,y::Symbolic) = Base.:^(x,y)
 # Simplification utilities
 function has_trig_exp(term)
     !istree(term) && return false
-    fns = (sin, cos, tan, cot, sec, csc, exp)
+    fns = (sin, cos, tan, cot, sec, csc, exp, cosh, sinh)
     op = operation(term)
 
-    if Base.@nany 7 i->fns[i] === op
+    if Base.@nany 9 i->fns[i] === op
         return true
     else
         return any(has_trig_exp, arguments(term))
diff --git a/test/rulesets.jl b/test/rulesets.jl
index cd1ada549..4730cc102 100644
--- a/test/rulesets.jl
+++ b/test/rulesets.jl
@@ -100,13 +100,25 @@ end
     @eqtest simplify(1 + y + cot(x)^2) == csc(x)^2 + y
     @eqtest simplify(cos(x)^2 - 1) == -sin(x)^2
     @eqtest simplify(sin(x)^2 - 1) == -cos(x)^2
+
+    @eqtest simplify(cosh(x)^2 + 1 - sinh(x)^2) == 2
+    @eqtest simplify(cosh(y)^2 + 1 - sinh(y)^2) == 2
+    @eqtest simplify(-sinh(y)^2 + cosh(y)^2 + 1) == 2
+
+    @eqtest simplify(cosh(x)^2 - 1) == sinh(x)^2
+    @eqtest simplify(sinh(x)^2 + 1) == cosh(x)^2
 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(r * sin(x / 2)^2 - r * cos(x / 2)^2) == -r * cos(x)
     @eqtest simplify(2cos(x) * sin(x)) == sin(2x)
+
+    @eqtest simplify(r * cosh(x / 2)^2 + r * sinh(x / 2)^2) == r * cosh(x)
+    @eqtest simplify(r * sinh(x / 2)^2 + r * cosh(x / 2)^2) == r * cosh(x)
+    @eqtest simplify(2cosh(x) * sinh(x)) == sinh(2x)
 end
 
 @testset "Exponentials" begin