Question

Coefficient of x^(3) in the expansion of tan x is
a 1/4
b 2/15
c 1/3
d 1/2​

Answer 1

Answer:

c 1/3.

Step-by-step explanation:

Correct option is c 1/3

Answer 2

Given:

Expansion of tan x

To Find:

coefficient of x³ in the expansion of tan x

Solution:

We know as per maclaurin series,

f(x) = f(0)+ x. f'(0)/1! + x². f''(0)/2! + x³ f'''(0)3!+..xⁿ.f⁽ⁿ⁾(0)n!+...

now put f(x) = tanx

ao,

⇒ f(x) = tan(x)

   f(0) = 0

⇒ f'(x) = sec²x

   f'(0) = 1

⇒ f''(x)=(2secx)(secxtanx)

           =2sec²x.tanx

           =2(1+tan2x).tanx

           =2(tanx+tan³x)

   f''(0) = 0

⇒ f'''(x) = 2{sec²x+3tan²xsec²x}

           =2sec²x{1+3tan²x}

           =2sec²x{1+3(sec²x−1)}

           =2sec²x{1+3sec²x−3}

           =6sec⁴x−4sec²x

      f'''(0) = 2

put this value in above equation,

⇒ f(x) = f(0)+ x. f'(0)/1! + x². f''(0)/2! + x³ f'''(0)3!+..xⁿ.f⁽ⁿ⁾(0)n!+...

⇒ tanx = 0 + x 1/1 + x². 0/2 + x³. 2/6 +....

⇒ tanx = x + 1/3 x³ +......

Hence, the cofficient of x³ is 1/3.