Prove that In square root 1-cos x/1+cos x=tan (x/2)
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Answer:
tan
−1
(
1+cosx
−
1−cosx
1+cosx
+
1−cosx
)
=tan
−1
(
(
1+cosx
)
2
−(
1−cosx
)
2
(
1+cosx
+
1−cosx
)
2
)
=tan
−1
(
1+cosx−1+cosx
1+cosx+1−cosx+2
1−cos
2
x
)
=tan
−1
(
2cosx
2+2
1−sin
2
x
)
=tan
−1
(
cosx
1+sinx
)
=tan
−1
⎝
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎛
1+tan
2
2
x
1−tan
2
2
x
1+
1+tan
2
2
x
2tan
2
x
⎠
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎞
=tan
−1
⎝
⎜
⎜
⎛
(1−tan
2
x
)
2
(1+tan
2
x
)
(1+tan
2
x
)
2
(1+tan
2
x
)
⎠
⎟
⎟
⎞
=tan
−1
⎝
⎛
1−tan
2
x
1+tan
2
x
⎠
⎞
=tan
−1
⎝
⎛
1−tan
4
x
tan
2
x
tan
4
π
+tan
2
x
⎠
⎞
As π<x<
2
3x
=tan
−1
(tan(
4
π
−
2
x
))
=
4
π
−
2
x
Hence, the answer is
4
π
−
2
x
.
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