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Byju's Answer
Standard XII
Mathematics
Trigonometric Ratios of Multiples of an Angle
If sinθ +si...
Question
If
sin
θ
+
sin
2
θ
=
1
,
then prove that
cos
12
θ
+
3
cos
10
θ
+
3
cos
8
θ
+
cos
6
θ
−
1
=
0
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Solution
The question should be if
sin
θ
+
sin
2
θ
=
1
then prove that
cos
12
θ
+
cos
6
θ
+
3
cos
10
θ
+
3
cos
8
θ
−
1
=
0
We have
sin
θ
+
sin
2
θ
=
1
⇒
sin
θ
=
1
−
sin
2
θ
⇒
sin
θ
=
cos
2
θ
..........
[
s
i
n
²
θ
+
c
o
s
²
θ
=
1
]
Squaring both sides, we get
⇒
sin
2
θ
=
cos
4
θ
⇒
1
−
cos
2
θ
=
cos
4
θ
......
[
s
i
n
²
θ
+
c
o
s
²
θ
=
1
]
⇒
cos
4
θ
+
cos
2
θ
=
1
Cubing both sides,we get
⇒
(
cos
4
θ
+
cos
2
θ
)
3
=
1
Using,
(
a
+
b
)
3
=
a
3
+
3
a
2
b
+
3
a
b
2
+
b
3
⇒
cos
12
θ
+
cos
6
θ
+
3
cos
6
θ
(
cos
4
θ
+
cos
2
θ
)
=
1
⇒
cos
12
θ
+
cos
6
θ
+
3
cos
10
θ
+
3
cos
8
θ
−
1
=
0
⇒
cos
12
θ
+
3
cos
10
θ
+
3
cos
8
θ
+
cos
6
θ
−
1
=
0
Hence proved.
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Similar questions
Q.
If
sin
θ
+
sin
2
θ
+
sin
3
θ
=
1
,
then
cos
6
θ
−
4
cos
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θ
+
8
cos
2
θ
=
.
.
.
.
.
.
.
Q.
Prove that:
1
csc
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cot
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csc
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+
cot
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Q.
Prove that
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sin
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−
cos
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θ
=
1
Q.
Prove that:
[
1
+
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tan
2
θ
]
[
1
+
1
cot
2
θ
]
=
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If
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+
.
.
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u
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