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Tardigrade
Question
Mathematics
If x = cosec[tan-1 cos(cot-1(sec(sin-1 a))) ] and y = sec [cot-1 sin(tan-1(cosec(cos-1 a))) ] where a ∈ [0, 1]. Find the relationship between x and y in terms of a.
Q. If
x
=
cosec
[
t
a
n
−
1
{
cos
(
co
t
−
1
(
sec
(
s
i
n
−
1
a
)
)
)
}
]
and
y
=
sec
[
co
t
−
1
{
s
in
(
t
a
n
−
1
(
cosec
(
co
s
−
1
a
)
)
)
}
]
where
a
∈
[
0
,
1
]
. Find the relationship between
x
and
y
in terms of
a
.
3777
195
Inverse Trigonometric Functions
Report Error
A
x
2
=
y
2
=
a
2
39%
B
x
2
=
y
2
=
3
+
a
2
17%
C
x
2
=
y
2
=
3
−
a
2
22%
D
none of these
22%
Solution:
We have,
x
=
cosec
[
t
a
n
−
1
{
cos
(
co
t
−
1
(
sec
(
s
i
n
−
1
a
)
)
)
}
]
=
cosec
[
t
a
n
−
1
{
cos
(
co
t
−
1
(
sec
(
se
c
−
1
1
−
a
2
1
)
)
)
}
]
=
cosec
[
t
a
n
−
1
{
cos
(
co
t
−
1
1
−
a
2
1
)
}
]
=
cosec
[
t
a
n
−
1
{
cos
(
co
s
−
1
2
−
a
2
1
)
}
]
=
cosec
(
t
a
n
−
1
2
−
a
2
1
)
=
cosec
(
cose
c
−
1
3
−
a
2
)
=
3
−
a
2
and,
y
=
sec
[
co
t
−
1
{
s
in
(
t
a
n
−
1
(
cosec
(
co
s
−
1
a
)
)
)
}
]
=
sec
[
co
t
−
1
{
s
in
(
t
a
n
−
1
(
cosec
(
cose
c
−
1
1
−
a
2
1
)
)
)
}
]
=
sec
[
co
t
−
1
{
s
in
(
t
a
n
−
1
1
−
a
2
1
)
}
]
=
sec
[
co
t
−
1
{
s
in
(
t
a
n
−
1
2
−
a
2
1
)
}
]
=
sec
(
co
t
−
1
2
−
a
2
1
)
=
sec
(
se
c
−
1
3
−
a
2
)
=
3
−
a
2
∴
x
2
=
y
2
=
3
−
a
2