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NIMCET 2012 Question Paper with solutions
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© examsnet.com
Question : 25
Total: 120
If cos (α + β) =
4
5
and sin (α - β) =
5
13
, 0 < α , β ,
π
4
, then tan (2α) is equal to
56
33
63
65
16
63
33
56
Validate
Solution:
Given, cos (α + β) =
4
5
and sin (α - β) =
5
13
where , 0 < α , β <
π
4
Using the identity
s
i
n
2
θ
+
c
o
s
2
θ
= 1
Now , sin (α + β) =
√
1
c
o
s
2
(
α
β
)
=
√
1
−
16
25
=
√
9
25
∴ sin (α+ β) =
3
5
and cos (α - β) =
√
1
−
s
i
n
2
(
α
−
β
)
=
√
1
−
25
169
=
√
144
168
∴ cos (α - β) =
12
13
Now, tan 2α = tan }(α + β) + (α- β)}
=
t
a
n
(
α
+
β
)
+
t
a
n
(
α
−
β
)
1
−
t
a
n
(
α
+
β
)
.
t
a
n
(
α
−
β
)
=
s
i
n
(
α
β
)
c
o
s
(
α
β
)
+
s
i
n
(
α
β
)
c
o
s
(
α
β
)
1
−
s
i
n
(
α
+
β
)
c
o
s
(
α
β
)
.
s
i
n
(
−
β
)
c
o
s
(
α
β
)
=
3
5
×
5
4
+
5
13
×
13
12
1
−
(
3
5
×
5
4
)
(
5
13
×
13
12
)
=
3
4
+
5
12
1
−
15
4.12
=
(
9
+
5
)
12
(
1
−
15
4
×
12
)
=
14
12
−
15
4
=
14
×
4
33
=
56
33
© examsnet.com
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