Resolved: "If tan⁡(nx) is expanded in powers of tan⁡(x)..."

Kale Mcclain

Answered question

2022-04-21

\tan (n x)

is expanded in powers of

\tan (x)

then what are the constant term and coefficient of

\tan (x)

in the expansion?

Answer & Explanation

tigging9k0

Beginner2022-04-22Added 19 answers

For real x and $n \in Z_{+}$ , we have
$\cos (n x) (1 + i \tan (n x)) = e^{\in x} = {(e^{i x})}^{n} = (\cos (x) {(1 + i \tan (x))}^{n}$
When we compare real and imaginary parts on both sides, we get
$\tan (n x) = \frac{I (L H S)}{R (L H S)} = \frac{I (R H S)}{R (R H S)} = \frac{I ({(1 + i \tan x)}^{n})}{R ({(1 + i \tan x)}^{n})}$
This leads to
$\tan (n x) = \frac{\sum_{k = 0}^{⌊ \frac{n - 1}{2} ⌋} (- 1)^{k} ((n), (2 k + 1)) \tan (x)^{2 k + 1}}{\sum_{k = 0}^{⌊ \frac{n}{2} ⌋} (- 1)^{k} ((n), (2 k)) \tan (x)^{2 k}} = \frac{n \tan (x) + O (\tan (x)^{3})}{1 + O (\tan (x)^{2})}$
$= 0 + n \tan (x) + O ({\tan (x)}^{3})$
Thus, the constant term is 0 and the coefficient in front of $\tan (x)$ is n.

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