If cosα=3/5 and cosβ=5/13,where α,β are acute angles.then prove that
Answers
Answer:
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Step-by-step explanation:
cosα=
5
3
⇒sinα=
5
4
and cosβ=
13
5
⇒sinβ=
13
12
Then
sinα+sinβ=
5
4
+
13
12
=
65
112
....(1)
cosα+cosβ=
5
3
+
13
5
=
65
64
....(2)
Adding and squaring (1) and (2), we get
sin
2
α+sin
2
β+2sinαsinβ+cos
2
α+cos
2
β+2cosαcosβ=
65
256
⇒2cos(α−β)=
65
256
−2=
65
126
⇒cos(α−β)=
65
63
A) Using cos2x=1−2sin
2
x⇒sin
2
x=
2
1−cos2x
We get
sin
2
(
2
α−β
)=
2
1−
65
63
=
65
1
B) Using cos2x=2cos
2
x−1⇒cos
2
x=
2
cos2x+1
cos
2
(
2
α−β
)=
65
64
C) tan
2
(
2
α−β
)=
cos
2
(
2
α−β
)
sin
2
(
2
α−β
)
=
65
64
65
1
=
64
1
D) sin(α+β)=sinαcos+cosαsinβ
=
5
4
×
13
5
+
5
3
×
13
12
=
65
56
Step-by-step explanation:
cosα=3/5 and cosβ=5/13,where α,β are acute angles.then prove that
Using cos2x=1−2sin2x⇒sin2x=21−cos2x
We get
sin2(2α−β)=21−6563=651
B) Using cos2x=2cos2x−1⇒cos2x=2cos2x+1
cos2(2α−β)=6564
C) tan2(2α−β)=cos2(2α−β)sin2(2α−β)=6564