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Byju's Answer
Standard XII
Mathematics
Derivative of Standard Functions
If acos3θ +...
Question
If
a
cos
3
θ
+
3
a
sin
2
θ
cos
θ
=
m
and
a
sin
3
θ
+
3
a
sin
θ
cos
2
θ
=
n
, find
(
m
+
n
)
2
/
3
+
(
m
−
n
)
2
/
3
=
A
2
a
2
/
3
.
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B
2
a
4
/
3
.
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C
a
2
/
3
.
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D
N
o
n
e
o
f
t
h
e
s
e
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Solution
The correct option is
A
2
a
2
/
3
.
We have
m
=
a
cos
3
θ
+
3
a
sin
2
θ
cos
θ
&
n
=
a
sin
3
θ
+
3
a
sin
θ
cos
2
θ
Now,
(
m
+
n
)
2
/
3
+
(
m
−
n
)
2
/
3
=
[
(
a
cos
3
θ
+
3
a
sin
2
θ
cos
θ
)
+
(
a
sin
3
θ
+
3
a
sin
θ
cos
2
θ
)
]
2
/
3
+
[
(
a
cos
3
θ
+
3
a
sin
2
θ
cos
θ
)
−
(
a
sin
3
θ
+
3
a
sin
θ
cos
2
θ
)
]
2
/
3
=
a
2
/
3
[
(
sin
3
θ
+
cos
3
θ
)
+
3
a
sin
θ
cos
θ
(
sin
θ
+
cos
θ
)
]
2
/
3
+
a
2
/
3
[
(
cos
3
θ
−
sin
3
θ
)
−
3
a
sin
θ
cos
θ
(
cos
θ
−
sin
θ
)
]
2
/
3
[Now as we know,
a
3
+
b
3
+
3
a
b
(
a
+
b
)
=
(
a
+
b
)
3
&
(
a
3
−
b
3
−
3
a
b
(
a
−
b
)
=
(
a
−
b
)
3
]
So,
=
a
2
/
3
{
[
(
cos
θ
+
sin
θ
)
3
]
2
/
3
+
[
(
cos
θ
−
sin
θ
)
3
]
2
/
3
}
=
a
2
/
3
[
(
cos
θ
+
sin
θ
)
2
+
(
cos
θ
−
sin
θ
)
2
]
=
a
2
/
3
(
cos
2
θ
+
2
sin
θ
cos
θ
+
sin
2
θ
+
cos
2
θ
−
2
sin
θ
cos
θ
+
sin
2
θ
)
=
a
2
/
3
(
1
+
1
)
=
2
a
2
/
3
So,
(
m
+
n
)
2
/
3
+
(
m
−
n
)
2
/
3
=
2
a
2
/
3
Suggest Corrections
1
Similar questions
Q.
If a cos
3
θ + 3a sin
2
θ cos θ = m and a sin
3
θ + 3a sin θ cos
2
θ = n, prove that
(m + n)
2/3
+ (m − n)
2/3
= 2a
2
/3
.
Q.
Prove the following trigonometric identities.
If a cos
3
θ + 3 a cos θ sin
2
θ = m, a sin
3
θ + 3 a cos
2
θ sin θ = n, prove that (m + n)
2/3
+ (m − n)
2/3
= 2a
2
/3