expand sin^6 theta interms of cos theta
Answers
Answer:
sin
6
(
x
)
=
−
1
32
(
cos
(
6
x
)
−
6
cos
(
4
x
)
+
15
cos
(
2
x
)
−
10
)
Step-by-step explanation:
We want to expand
sin
6
(
x
)
One way is to use these identities repeatedly
sin
2
(
x
)
=
1
2
(
1
−
cos
(
2
x
)
)
cos
2
(
x
)
=
1
2
(
1
+
cos
(
2
x
)
)
This often gets quite long, which sometimes leads to mistake
Another way is to use the complex numbers (and Euler's formula)
We can express sine and cosine as
sin
(
x
)
=
e
i
x
−
e
−
i
x
2
i
and
cos
(
x
)
=
e
i
x
+
e
−
i
x
2
Thus
sin
6
(
x
)
=
(
e
i
x
−
e
−
i
x
2
i
)
6
=
(
e
i
x
−
e
−
i
x
)
6
−
64
=
e
6
i
x
−
6
e
4
i
x
+
15
e
2
i
x
−
20
+
15
e
−
2
i
x
−
6
e
−
4
i
x
+
e
−
6
i
x
−
64
=
−
1
32
e
6
i
x
+
e
−
6
i
x
−
6
e
4
i
x
−
6
e
−
4
i
x
+
15
e
2
i
x
+
15
e
−
2
i
x
−
20
2
=
−
1
32
(
cos
(
6
x
)
−
6
cos
(
4
x
)
+
15
cos
(
2
x
)
−
10
)
The third way is using De Moivre's theorem, we can express
2
cos
(
n
x
)
=
z
n
+
1
z
n
and
2
i
sin
(
n
x
)
=
z
n
−
1
z
n
where
z
n
=
(
cos
(
x
)
+
i
sin
(
x
)
)
n
=
cos
(
n
x
)
+
i
sin
(
n
x
)
Thus
(
2
i
sin
(
x
)
)
6
=
(
z
−
1
z
)
6
⇒
sin
6
(
x
)
=
−
1
64
(
z
−
1
z
)
6
Expand the binomial on the right hand side
B
I
N
=
(
z
6
+
1
z
6
−
6
z
4
−
6
z
4
+
15
z
2
+
15
z
2
−
20
)
R
H
S
=
(
z
6
+
1
z
6
−
6
⋅
(
z
4
+
1
z
4
)
+
15
⋅
(
z
+
1
z
)
+
20
)
R
H
S
=
(
2
cos
(
6
x
)
−
12
cos
(
4
x
)
+
30
cos
(
2
x
)
−
20
)
Thus
sin
6
(
x
)
=
−
1
64
(
2
cos
(
6
x
)
−
12
cos
(
4
x
)
+
30
cos
(
2
x
)
−
20
)
sin
6
(
x
)
=
−
1
32
(
cos
(
6
x
)
−
6
cos
(
4
x
)
+
15
cos
(
2
x
)
−
10
)