Grade 10Trigonometry
Proof that Cos2x+asin2x=cosx-sinx and find the value of a
Proof that Cos2x+asin2x=cosx-sinx and find the value of a
1 Answer
Varun
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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