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CBSE Class 12 Math 2011 Solved Paper

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Question : 12
Total: 29
Prove the following :
cot−1[
√1+sinx+√1−sinx
√1+sinx−√1−sinx
] =
x
2
 , x ∊ (0,
π
2
)
OR
Find the value of tan−1(
x
y
) - tan−1(
x−y
x+y
)
Solution:  
cot−1[
√1+sinx+√1−sinx
√1+sinx−√1−sinx
]
=
cot−1[
√sin2(
x
2
)+cos2(
x
2
)+sin2(
x
2
)+√sin2(
x
2
)+cos2(
x
2
)−sin2(
x
2
)
√sin2(
x
2
)+cos2(
x
2
)+sin2(
x
2
)−√sin2(
x
2
)+cos2(
x
2
)−sin2(
x
2
)
]

[Since, sin2 A + cos2 A = 1]
=
cot−1[
√sin2(
x
2
)+cos2(
x
2
)+2sin(
x
2
)cos(
x
2
)+√sin2(
x
2
)+cos2(
x
2
)−2sin(
x
2
)cos(
x
2
)
√sin2(
x
2
)+cos2(
x
2
)+2sin(
x
2
)cos(
x
2
)−√sin2(
x
2
)+cos2(
x
2
)−2sin(
x
2
)cos(
x
2
)
]
[Since, sin2A = 2 sinA cosA]
=
cot−1[
√(cos
x
2
+sin
x
2
)2+√(cos
x
2
−sin
x
2
)2
√(cos
x
2
+sin
x
2
)2−√(cos
x
2
−sin
x
2
)2
]

= cot−1(
2cos
x
2
2sin
x
2
) = cot−1(cot
x
2
)
=
x
2
Hence proved.
OR
tan−1(
x
y
) - tan−1(
x−y
x+y
)
tan−1(
x
y
) - tan−1(
x
y
−1
x
y
+1
)
= tan−1(
x
y
) - tan−1(
x
y
−1
1+
x
y
)
tan−1(
x
y
) - [tan−1(
x
y
)−tan−1(1)] [Since tan−1 a - tan−1 b = tan−1(
a−b
1+ab
)]
tan−1(
x
y
) - tan−1(
x
y
) + tan−1(1)
= tan−1(1) =
π
4
Thus tan−1(
x
y
) - tan−1(
x−y
x+y
) =
π
4
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