Byju's Answer
Standard XIII
Mathematics
Product of Trigonometric Ratios in Terms of Their Sum
If in a ABC, ...
Question
If in a
△
A
B
C
,
tan
A
2
,
tan
B
2
,
tan
C
2
are in H.P., then the minimum value of
cot
B
2
is
A
3
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B
2
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C
√
3
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D
√
2
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Solution
The correct option is
C
√
3
In a
△
A
B
C
,
cot
A
2
+
cot
B
2
+
cot
C
2
=
cot
A
2
cot
B
2
cot
C
2
…
(
1
)
∵
tan
A
2
,
tan
B
2
,
tan
C
2
are in H.P.
∴
cot
A
2
,
cot
B
2
,
cot
C
2
are in A.P.
⇒
2
cot
B
2
=
cot
A
2
+
cot
C
2
From
(
1
)
,
3
cot
B
2
=
cot
A
2
cot
B
2
cot
C
2
⇒
cot
A
2
cot
C
2
=
3
Now,
A
.
M
.
≥
G
.
M
.
⇒
cot
A
2
+
cot
C
2
≥
2
√
cot
A
2
cot
C
2
⇒
2
cot
B
2
≥
2
√
3
∴
cot
B
2
≥
√
3
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Similar questions
Q.
If in a
△
A
B
C
,
tan
A
2
,
tan
B
2
,
tan
C
2
are in H.P., then the minimum value of
cot
B
2
is
Q.
In
△
A
B
C
, if
tan
A
2
and
tan
A
2
are the roots of the equation
6
x
2
−
5
x
+
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=
0
, then
Q.
If ABC is a triangle and
tan
A
2
,
tan
B
2
,
tan
C
2
are in H.P., then find the minimum value of
cot
B
/
2
.
Q.
If in a
△
A
B
C
,
tan
A
2
,
tan
B
2
,
tan
C
2
are in H.P., then the minimum value of
cot
B
2
is
Q.
In triangle ABC, prove the following:
c
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=
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+
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tan
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