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Byju's Answer
Standard XI
Mathematics
Trigonometric Equations
If sin x + co...
Question
If
sin
x
+
cos
x
=
√
y
2
+
1
y
2
for
x
∈
[
0
,
π
]
has a solution, then which of the following is/are correct?
A
x
=
π
4
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B
y
=
−
1
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C
y
=
1
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D
x
=
3
π
4
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Solution
The correct option is
C
y
=
1
R.H.S.=
y
2
+
1
y
2
≥
2
√
y
2
×
1
y
2
⇒
y
2
+
1
y
2
≥
2
⇒
√
y
2
+
1
y
2
≥
√
2
L
.
H
.
S
.
=
sin
x
+
cos
x
=
√
2
sin
(
π
4
+
x
)
⇒
sin
x
+
cos
x
≤
√
2
So, the equality holds only when
y
2
+
1
y
2
=
2
⇒
y
=
±
1
and
sin
x
+
cos
x
=
√
2
⇒
sin
(
x
+
π
4
)
=
1
⇒
x
+
π
4
=
π
2
,
5
π
2
⇒
x
=
π
4
,
9
π
4
,
⋯
∴
x
=
π
4
Suggest Corrections
0
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