the coefficient of x^2 in the expansion of the cos 2x
Answers
Answer:
Correct option is
C
45
2
We have to find the coefficient of third term in Maclaurin series of sin
2
x.
The Maclaurin series is given by f(x)=∑
k=0
∞
k!
f
(k)
(a)
x
k
where a=0
We have f(x)=sin
2
x
Since we have to find the coefficient of the third term, let us take n=8.
∴f(x)≈∑
k=0
8
k!
f
(k)
(0)
x
k
f
(0)
(x)=sin
2
x,⇒f
(0)
(0)=0
f
(1)
(x)=2sinxcosx,⇒f
(1)
(0)=0
f
(2)
(x)=−2sin
2
x+2cos
2
x,⇒f
(2)
(0)=2
f
(3)
(x)=−8cosxsinx,⇒f
(3)
(0)=0
f
(4)
(x)=8sin
2
x−8cos
2
x,⇒f
(4)
(0)=−8
f
(5)
(x)=32sinxcosx,⇒f
(5)
(0)=0
f
(6)
(x)=−32sin
2
x+32cos
2
x,⇒f
(6)
(0)=32
f
(7)
(x)=−128sinxcosx,⇒f
(7)
(0)=0
f
(8)
(x)=128sin
2
x−128cos
2
x,⇒f
(8)
(0)=−128
∴f(x)≈0x
0
+0x
1
+
2!
2
x
2
+
3!
0
x
3
+
4!
−8
x
4
+
5!
0
x
5
+
6!
32
x
6
+
7!
0
x
7
+
8!
−128
x
8
⇒f(x)≈x
2
−
3
1
x
4
+
45
2
x
6
−
135
1
x
5
Thus the coefficient of third term is
45
2