If cos x+cos y=a and sin x+sin y=b then show that sin 2x+sin 2y=2ab(1-2/a^2+b^2)
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Answer:
Given that,
→
cos
x
+
c
o
y
=
a
....[1]
→
sin
x
+
sin
y
=
b
....[2]
Squaring and adding [1] and [2], we get,
→
cos
2
x
+
2
cos
x
cos
y
+
cos
2
y
+
sin
2
x
+
2
sin
x
sin
y
+
sin
2
y
=
a
2
+
b
2
→
2
+
2
(
cos
x
cos
y
+
sin
x
sin
y
)
=
a
2
+
b
2
→
2
(
1
+
cos
(
x
−
y
)
)
=
a
2
+
b
2
→
cos
(
x
−
y
)
=
a
2
+
b
2
2
−
1
Dividing equation [1] by [2], we get,
→
cos
x
+
cos
y
sin
x
+
sin
y
=
a
b
→
2
cos
(
x
+
y
2
)
cos
(
x
−
y
2
)
2
sin
(
x
+
y
2
)
cos
(
x
−
y
2
)
=
a
b
→
cot
(
x
+
y
2
)
=
a
b
→
tan
(
x
+
y
2
)
=
b
a
→
x
+
y
2
=
tan
−
1
(
b
a
)
→
x
+
y
=
2
tan
−
1
(
b
a
)
As,
2
tan
−
1
x
=
sin
−
1
(
2
x
1
+
x
2
)
,we have,
→
x
+
y
=
sin
−
1
⎛
⎜
⎜
⎝
2
⋅
(
b
a
)
1
+
(
b
a
)
2
⎞
⎟
⎟
⎠
=
sin
−
1
(
2
a
b
a
2
+
b
2
)
→
sin
(
x
+
y
)
=
2
a
b
a
2
+
b
2
Now,
L
H
S
=
sin
2
x
+
sin
2
y
=
2
sin
(
x
+
y
)
⋅
cos
(
x
−
y
)
=
2
[
2
a
b
a
2
+
b
2
]
[
a
2
+
b
2
2
−
1
]
=
2
a
b
[
2
a
2
+
b
2
⋅
a
2
+
b
2
2
−
2
a
2
+
b
2
]
=
2
a
b
[
1
−
2
a
2
+
b
2
]
=
R
H
S
Step-by-step explanation:
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