The following identities hold for all complex numbers zz and w.w.
sin
(
z
+
w
)
=
sin
(
z
)
cos
(
w
)
+
cos
(
z
)
sin
(
w
)
.
sin
(
z
+
w
)
=
sin
(
z
)
cos
(
w
)
+
cos
(
z
)
sin
(
w
)
.
(1)
cos
(
z
+
w
)
=
cos
(
z
)
cos
(
w
)
-
sin
(
z
)
sin
(
w
)
.
cos
(
z
+
w
)
=
cos
(
z
)
cos
(
w
)
-
sin
(
z
)
sin
(
w
)
.
(2)
sinh
(
z
+
w
)
=
sinh
(
z
)
cosh
(
w
)
+
cosh
(
z
)
sinh
(
w
)
.
sinh
(
z
+
w
)
=
sinh
(
z
)
cosh
(
w
)
+
cosh
(
z
)
sinh
(
w
)
.
(3)
cosh
(
z
+
w
)
=
cosh
(
z
)
cosh
(
w
)
+
sinh
(
z
)
sinh
(
w
)
.
cosh
(
z
+
w
)
=
cosh
(
z
)
cosh
(
w
)
+
sinh
(
z
)
sinh
(
w
)
.
(4)
We derive the first formula and leave
the others to an exercise.
First, for any two real numbers xx and y,y, we have
cos
(
x
+
y
)
+
i
sin
(
x
+
y
)
=
e
i
(
x
+
y
)
=
e
i
x
e
i
y
=
(
cos
x
+
i
sin
x
)
×
(
cos
y
+
i
sin
y
)
=
cos
x
cos
y
-
sin
x
sin
y
+
i
(
cos
x
sin
y
+
sin
x
cos
y
)
,
cos
(
x
+
y
)
+
i
sin
(
x
+
y
)
=
e
i
(
x
+
y
)
=
e
i
x
e
i
y
=
(
cos
x
+
i
sin
x
)
×
(
cos
y
+
i
sin
y
)
=
cos
x
cos
y
-
sin
x
sin
y
+
i
(
cos
x
sin
y
+
sin
x
cos
y
)
,
(5)which, equating real and imaginary parts, gives that
cos
(
x
+
y
)
=
cos
x
cos
y
-
sin
x
sin
y
cos
(
x
+
y
)
=
cos
x
cos
y
-
sin
x
sin
y
(6)and
sin
(
x
+
y
)
=
sin
x
cos
y
+
cos
x
sin
y
.
sin
(
x
+
y
)
=
sin
x
cos
y
+
cos
x
sin
y
.
(7)The second of these equations is exactly what we want,
but this calculation only shows that it holds for real numbers xx and y.y.
We can use the Identity Theorem to show that in fact this formula holds for all complex numbers zz and w.w.
Thus, fix a real number y.y.
Let f(z)=sinzcosy+coszsiny,f(z)=sinzcosy+coszsiny,
and let
g
(
z
)
=
sin
(
z
+
y
)
=
1
2
i
(
e
i
(
z
+
y
)
-
e
-
i
(
z
+
y
)
=
1
2
i
(
e
i
z
e
i
y
-
e
-
i
z
e
-
i
y
)
.
g
(
z
)
=
sin
(
z
+
y
)
=
1
2
i
(
e
i
(
z
+
y
)
-
e
-
i
(
z
+
y
)
=
1
2
i
(
e
i
z
e
i
y
-
e
-
i
z
e
-
i
y
)
.
(8)Then both ff and gg are power series functions of the variable z.z. Furthermore, by the previous calculation,
f(1/k)=g(1/k)f(1/k)=g(1/k) for all positive integers k.k.
Hence, by the Identity Theorem, f(z)=g(z)f(z)=g(z) for all complex z.z.
Hence we have the formula we want for all complex numbers zz and all real numbers y.y.
To finish the proof, we do the same trick one more time.
Fix a complex number z.z.
Let f(w)=sinzcosw+coszsinw,f(w)=sinzcosw+coszsinw,
and let
g
(
w
)
=
sin
(
z
+
w
)
=
1
2
i
(
e
i
(
z
+
w
)
-
e
-
i
(
z
+
w
)
=
1
2
i
(
e
i
z
e
i
w
-
e
-
i
z
e
-
i
w
)
.
g
(
w
)
=
sin
(
z
+
w
)
=
1
2
i
(
e
i
(
z
+
w
)
-
e
-
i
(
z
+
w
)
=
1
2
i
(
e
i
z
e
i
w
-
e
-
i
z
e
-
i
w
)
.
(9)Again, both ff and gg are power series functions of the variable w,w,
and they agree on the sequence {1/k}.{1/k}. Hence they agree everywhere,
and this completes the proof of the first addition formula.
