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Proof that Cos2x+asin2x=cosx-sinx and find the value of a

Proof that Cos2x+asin2x=cosx-sinx and find the value of a
 

Grade:10

1 Answers

Varun
28 Points
4 years ago
Explanation:
cos
x
+
sin
x
=
cos
2
x
+
sin
2
x
 .
 
∴
cos
x
−
cos
2
x
=
sin
2
x
−
sin
x
 .
 
∴
−
2
sin
(
x
+
2
x
2
)
sin
(
x
−
2
x
2
)
=
2
cos
(
2
x
+
x
2
)
sin
(
2
x
−
x
2
)
.
 
∴
+
sin
(
3
2
x
)
sin
(
+
1
2
x
)
=
cos
(
3
2
x
)
sin
(
1
2
x
)
.
 
∴
sin
(
3
2
x
)
sin
(
1
2
x
)
−
cos
(
3
2
x
)
sin
(
1
2
x
)
=
0
.
 
∴
sin
(
1
2
x
)
[
sin
(
3
2
x
)
−
cos
(
3
2
x
)
]
=
0
 .
 
∴
sin
(
1
2
x
)
=
0
,
or
,
sin
(
3
2
x
)
=
cos
(
3
2
x
)
.
 
Case 1 :  
sin
(
1
2
x
)
=
0
 .
 
sin
(
1
2
x
)
=
0
⇒
1
2
x
=
k
π
⇒
x
=
2
k
π
,
k
∈
Z
.
 
Case 2 :  
sin
(
3
2
x
)
=
cos
(
3
2
x
)
 .
 
Note that  
cos
(
3
2
x
)
 can not be 
0
,
  because, in that case, by the
 
virtue of the eqn.,  
sin
(
3
2
x
)
 will also be 
0
,
 contradicting,
 
sin
2
(
3
2
x
)
+
cos
2
(
3
2
x
)
=
1
 .
 
So, dividing by  
cos
(
3
2
x
)
≠
0
 , we get,
 
tan
(
3
2
x
)
=
1
=
tan
(
π
4
)
,
 giving, 
 
 
3
2
x
=
k
π
+
π
4
⇒
x
=
2
3
k
π
+
π
6
=
(
4
k
+
1
)
π
6
,
k
∈
Z
.
 
Altogether, The Soln. Set = 
{
2
k
π
}
∪
{
(
4
k
+
1
)
π
6
}
,
k
∈
Z
 .

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