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KEAM 2013 Math Paper
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© examsnet.com
Question : 42
Total: 120
If
c
o
s
x
+
c
o
s
2
x
=
1
,
then the value of
s
i
n
4
x
+
s
i
n
6
x
is equal to
−
1
+
√
5
−
1
−
√
5
2
1
−
√
5
2
1
+
√
5
2
1
−
√
5
Validate
Solution:
Given
cos
x
+
cos
2
x
=
1
....(i)
⇒
cos
2
x
+
cos
x
−
1
=
0
∴
cos
x
=
−
1
±
√
1
+
4
2
( ∵ quadratic in
cos
x
)
=
−
1
±
√
5
2
=
−
1
+
√
5
2
(
∵
cos
x
≠
−
1
−
√
5
2
)
Also from Eq. (i).
cos
x
=
1
−
cos
2
x
⇒
cos
x
=
sin
2
x
....(ii)
Now,
sin
4
x
+
sin
6
x
=
cos
2
x
+
cos
3
x
=
cos
2
x
(
1
+
cos
x
)
=
(
−
1
+
√
5
2
)
2
(
1
+
−
1
+
√
5
2
)
=
(
−
1
+
√
5
2
)
2
(
1
+
√
5
2
)
=
(
−
1
+
√
5
2
)
(
5
−
1
4
)
=
(
−
1
+
√
5
2
)
×
1
=
−
1
+
√
5
2
© examsnet.com
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