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CBSE Class 12 Math 2008 Solved Paper

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Question : 12
Total: 29
Prove the following:
tan−1
1
3
+tan−1
1
5
 + tan−1
1
7
+tan−1
1
8
 =
π
8
Solution:  
L.H.S. = tan−1
1
3
+tan−1
1
5
 + tan−1
1
7
+tan−1
1
8
= tan−1|
1
3
+
1
5
1−(
1
3
)(
1
5
)
| + tan−1|
1
7
+
1
8
1−(
1
7
)(
1
8
)
|
= tan−1|
5+3
15
15−1
15
| + tan−1|
8+7
56
56−1
56
|
= tan−1|
8
15
14
15
| + tan−1|
15
56
55
56
|
= tan−1(
8
14
) + tan−1(
15
55
)
= tan−1(
4
7
)+tan−1(
3
11
)
= tan−1(
4
7
+
3
11
1−(
4
7
)(
3
11
)
)
= tan−1(
44+21
77
77−12
77
)
= tan−1(
65
65
)
= tan−1 1
= tan−1(tan
π
4
)
=
π
4
= R.H.S.
Hence proved.
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