Show that for a complex number z
(
z
+
1
z
)
=
2
c
o
s
θ
W
e
k
n
o
w
t
h
a
t
z
=
(
c
o
s
θ
+
i
s
i
n
θ
)
a
n
d
t
h
a
t
1
z
=
(
c
o
s
θ
+
i
s
i
n
θ
)
−
1
A
l
s
o
,
b
y
d
e
M
o
i
v
r
e
′
s
T
h
e
o
r
e
m
(
c
o
s
θ
+
i
s
i
n
θ
)
−
1
=
(
c
o
s
(
−
θ
)
+
i
s
i
n
(
−
θ
)
)
T
h
e
r
e
f
o
r
e
(
z
+
1
z
)
=
(
c
o
s
θ
+
i
s
i
n
θ
)
+
(
c
o
s
(
−
θ
)
+
i
s
i
n
(
−
θ
)
)
A
n
d
b
e
c
a
u
s
e
c
o
s
(
−
θ
)
=
c
o
s
(
θ
)
a
n
d
s
i
n
(
−
θ
)
=
−
s
i
n
(
θ
)
)
(
z
+
1
z
)
=
(
c
o
s
θ
+
i
s
i
n
θ
)
+
(
c
o
s
(
θ
)
−
i
s
i
n
(
θ
)
)
=
2
c
o
s
θ
Show that for a complex number z
(
z
−
1
z
)
=
2
i
s
i
n
θ
W
e
k
n
o
w
t
h
a
t
z
=
(
c
o
s
θ
+
i
s
i
n
θ
)
a
n
d
t
h
a
t
1
z
=
(
c
o
s
θ
+
i
s
i
n
θ
)
−
1
A
l
s
o
,
b
y
d
e
M
o
i
v
r
e
′
s
T
h
e
o
r
e
m
(
c
o
s
θ
+
i
s
i
n
θ
)
−
1
=
(
c
o
s
(
−
θ
)
+
i
s
i
n
(
−
θ
)
)
T
h
e
r
e
f
o
r
e
(
z
−
1
z
)
=
(
c
o
s
θ
+
i
s
i
n
θ
)
−
(
c
o
s
(
−
θ
)
+
i
s
i
n
(
−
θ
)
)
A
n
d
b
e
c
a
u
s
e
c
o
s
(
−
θ
)
=
c
o
s
(
θ
)
a
n
d
s
i
n
(
−
θ
)
=
−
s
i
n
(
θ
)
)
(
z
−
1
z
)
=
(
c
o
s
θ
+
i
s
i
n
θ
)
−
(
c
o
s
(
θ
)
−
i
s
i
n
(
θ
)
)
=
2
i
s
i
n
θ