general solution of Cos2x+Cosx-2
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Answer:
x
=
2
n
π
for
∀
n
∈
Z
explanation:
cos
(
2
x
)
=
2
cos
2
(
x
)
−
1
XXXX
(This is one version of the double angle formula for cos)
So
XXXX
cos
(
2
x
)
+
cos
(
x
)
−
2
=
0
can be written as
XXXX
2
cos
2
(
x
)
+
cos
(
x
)
−
3
=
0
This can be factored as:
XXXX
(
2
cos
(
x
)
+
3
)
(
cos
(
x
)
−
1
)
=
0
So
cos
(
x
)
=
−
3
2
XXXX
or
XXXX
cos
(
x
)
=
1
But
−
3
2
is not within the range of
cos
(
x
)
for any value of
(
x
)
.
XXXX
(
cos
(
x
)
∈
[
−
1
,
+
1
]
)
So
cos
(
x
)
=
1
XXXX
within the range
x
∈
[
0
,
2
π
)
x
=
0
or, in general
x
=
2
n
π
for
∀
n
∈
Z
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