Question

Prove that `tan_1 x=x-x^3/3+x^5/5+.............`

Answer 1

Let y=` tan_1 x`

diff. W.r.t.x, 

∴ `y_1=1/(x^2+1)`

Series expansion of 𝒚𝟏 ,
We know that , 

`1/(1+x)=1-x+x^2-x^3+........` 

∴ `y_1=1-x^2+x^4-x^5` 

Integrate 𝒚𝟏 to find series expansion of y, 

∴ `y=int(1-x^2+x^4-x^5+.........)dx` 

∴ `y=x- x^3/3+x^5/5............` 

Hence Proved .