At the risk of piling on after correct answers have already been given, I’ll offer another way to address this problem. Recall that, if
θ
is real, cos()=Re()
cos
(
θ
)
=
R
e
(
e
i
θ
)
. So, with
A
and
B
real,
(+)+(−)=(+−)=2cos()
e
i
(
A
+
B
)
+
e
i
(
A
−
B
)
=
e
i
A
(
e
i
B
+
e
−
i
B
)
=
2
e
i
A
cos
(
B
)
Finally,
cos(+)+cos(−)=Re((+)+(−))=Re(2cos())
cos
(
A
+
B
)
+
cos
(
A
−
B
)
=
R
e
(
e
i
(
A
+
B
)
+
e
i
(
A
−
B
)
)
=
R
e
(
2
e
i
A
cos
(
B
)
)
=2cos()cos()