Question

The constant term in the expansion of tan-'x in powers of (x - 1) is​

Answer 1

Answer:

x

please mark me as brainlist

Answer 2

π/4

Expansion:

To find the constant term in the expansion of tan-'x in powers of (x - 1) we will use Taylor's theorem.

y(x) = y(1) + (x-1)$y_{1}$(1) + $\frac{(x-1)^{2}}{2!}$ $y_{2}$  ____ eq (1)

y(x) = tan$.^{-1}$x

⇒ y(1) = tan$.^{-1}$(1) = $\frac{\pi }{4}$

$y_{1}$(x) = $\frac{1}{1+x^{2} }$

y(1) = 1/2

$y_{2} (1)$ = -1/2

Now putting them in eq (1)

y(x) = $\frac{\pi }{4}$ + (x-1)1/2 + $\frac{(x-1)^{2}}{2!}$ (-1/2)

y(x) = $\frac{\pi }{4}$ + 1/2 ((x-1) - $\frac{(x-1)^{2}}{2}$)

Hence, the constand tern in the expansion of tan-'x in powers of (x - 1) is π/4