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Byju's Answer
Standard XIII
Mathematics
AM,GM,HM Inequality
If sin4α +4co...
Question
If
sin
4
α
+
4
cos
4
β
+
2
=
4
√
2
sin
α
cos
β
;
α
,
β
∈
[
0
,
π
]
, then
cos
(
α
+
β
)
−
cos
(
α
−
β
)
is equal to :
A
√
2
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B
−
√
2
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C
0
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D
−
1
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Solution
The correct option is
B
−
√
2
sin
4
α
+
4
cos
4
β
+
2
=
4
√
2
sin
α
cos
β
Applying
A
.
M
≥
G
.
M
.
on LHS
sin
4
α
+
4
cos
4
β
+
1
+
1
4
≥
4
√
sin
4
α
×
4
cos
4
β
×
1
×
1
⇒
sin
4
α
+
4
cos
4
α
+
2
≥
4
√
2
sin
α
cos
β
But given that
sin
4
α
+
4
cos
4
β
+
2
=
4
√
2
sin
α
cos
β
This is possible only if
sin
4
α
=
4
cos
4
β
=
1
∴
sin
α
=
±
1
and
cos
β
=
±
1
√
2
⇒
sin
α
=
1
and
cos
β
=
±
1
√
2
(
∵
α
,
β
∈
[
0
,
π
]
)
∴
sin
β
=
1
√
2
cos
(
α
+
β
)
−
cos
(
α
−
β
)
=
−
2
sin
α
sin
β
=
−
√
2
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0
Similar questions
Q.
If
sin
4
α
+
4
cos
4
β
+
2
=
4
√
2
sin
α
cos
β
;
α
,
β
∈
[
0
,
π
]
, then
cos
(
α
+
β
)
−
cos
(
α
−
β
)
is equal to :
Q.
If
sin
4
α
+
4
cos
4
β
+
2
=
4
√
2
sin
α
cos
β
;
α
,
β
ϵ
[
0
,
π
]
, then
cos
(
α
+
β
)
−
cos
(
α
+
β
)
is equal to :
Q.
If
0
<
β
<
α
<
π
4
,
cos
(
α
+
β
)
=
3
5
and
cos
(
α
−
β
)
=
4
5
then
sin
2
α
is equal to
Q.
If
0
<
α
,
β
<
π
and
c
o
s
α
+
c
o
s
β
−
c
o
s
(
α
+
β
)
=
3
/
2
then
984
√
3
s
i
n
α
+
896
c
o
s
α
is equal to
Q.
If
0
<
α
,
β
<
π
and
cos
α
+
cos
β
−
cos
(
α
+
β
)
=
3
2
, then
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