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Byju's Answer
Standard XI
Mathematics
Euler's Representation
If tanα+iβ=ei...
Question
If
tan
(
α
+
i
β
)
=
e
i
θ
;
where
α
,
β
∈
R
,
θ
≠
(
2
n
+
1
)
π
2
,
n
∈
Z
and
i
=
√
−
1
, then
A
α
is an odd multiple of
π
2
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B
α
is an even multiple of
π
2
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C
α
is an odd multiple of
π
4
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D
α
is an even multiple of
π
4
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Solution
The correct option is
C
α
is an odd multiple of
π
4
tan
(
α
+
i
β
)
=
e
i
θ
⇒
tan
(
α
+
i
β
)
=
cos
θ
+
i
sin
θ
Taking complex conjugate, we get
tan
(
α
−
i
β
)
=
cos
θ
−
i
sin
θ
tan
2
α
=
tan
[
(
α
+
i
β
)
+
(
α
−
i
β
)
]
⇒
tan
2
α
=
tan
(
α
+
i
β
)
+
tan
(
α
−
i
β
)
1
−
tan
(
α
+
i
β
)
tan
(
α
−
i
β
)
⇒
1
tan
2
α
=
1
−
tan
(
α
+
i
β
)
tan
(
α
−
i
β
)
tan
(
α
+
i
β
)
+
tan
(
α
−
i
β
)
⇒
cot
2
α
=
1
−
(
cos
2
θ
+
sin
2
θ
)
2
cos
θ
⇒
cot
2
α
=
0
[
∵
θ
≠
(
2
n
+
1
)
π
2
]
⇒
2
α
=
(
2
m
+
1
)
π
2
,
m
∈
Z
⇒
α
=
(
2
m
+
1
)
π
4
,
m
∈
Z
Suggest Corrections
1
Similar questions
Q.
If
tan
(
α
+
i
β
)
=
e
i
θ
;
where
α
,
β
∈
R
,
θ
≠
(
2
n
+
1
)
π
2
,
n
∈
Z
and
i
=
√
−
1
, then
Q.
If
tan
α
−
tan
β
=
m
and
cot
α
−
cot
β
=
n
, then prove that
cot
(
α
−
β
)
=
1
m
−
1
n
Q.
Let
f
:
R
→
R
be an invertible and a differentiable function defined by
f
(
x
)
=
{
x
2
+
a
x
−
6
,
x
≤
2
α
x
2
+
β
,
x
>
2
where
a
,
β
∈
Z
,
α
∈
R
and
a
≥
−
5.
If
f
−
1
denotes the inverse function of
f
,
then
Q.
If
α
and
β
are real then
∣
∣
∣
α
+
i
β
β
+
i
α
∣
∣
∣
=
Q.
If
sin
(
α
+
β
)
=
1
and
sin
(
α
−
β
)
=
1
2
where
0
≤
α
,
β
≤
π
2
then find the value of
tan
(
α
+
2
β
)
tan
(
2
α
+
β
)
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