If cosθ=sin(2θ)

=0, what is the value of cos
4
θ+sin
4
θ+cos
3
θ+sin
3
θ+sin
2
θ+cos
2
θ+sinθ+cosθ?

Given:
cosθ=sin(2θ)⟹cosθ=2sinθcosθ
Since cosθ

=0, we divide both sides by cosθ:

1=2sinθ⟹sinθ=
2
1
​

For acute angles, θ=30
∘
.
Let's find the value of cosθ:

cosθ=cos30
∘
=
2
3

​

​

Now substitute sinθ=
2
1
​
and cosθ=
2
3

​

​
into the given expression:

E=(cos
4
θ+sin
4
θ)+(cos
3
θ+sin
3
θ)+(cos
2
θ+sin
2
θ)+(cosθ+sinθ)
Calculate each group step-by-step:

cos
2
θ+sin
2
θ=1

cosθ+sinθ=
2
3

​

​
+
2
1
​
=
2
3

​
+1
​


cos
3
θ+sin
3
θ=(
2
3

​

​
)
3
+(
2
1
​
)
3
=
8
3
3

​

​
+
8
1
​
=
8
3
3

​
+1
​


cos
4
θ+sin
4
θ=(
2
3

​

​
)
4
+(
2
1
​
)
4
=
16
9
​
+
16
1
​
=
16
10
​
=
8
5
​


Summing all components:

E=
8
5
​
+
8
3
3

​
+1
​
+1+
2
3

​
+1
​

Combine with a common denominator of 8:

E=
8
5+(3
3

​
+1)+8+4(
3

​
+1)
​

E=
8
5+3
3

​
+1+8+4
3

​
+4
​

E=
8
(5+1+8+4)+(3
3

​
+4
3

​
)
​

E=
8
18+7
3

​

​