# tan reduction formula

• Jul 25th 2010, 02:01 PM
Tclack
tan reduction formula
So the proof for the reduction formula for [tan(x)]^n I found here, and it makes sense

But I did this and it is very close, but not the same formula, what's wrong?

$\displaystyle \int tan^nx dx = \int tan^n^-^2x *tan^2x dx$
$\displaystyle =\int tan^n^-^2x(sec^2x -1) dx$
$\displaystyle = \int tan^n^-^2x*sec^2x dx - \int tan^n^-^2x dx$

here I did an integration by parts of the first integral above:

$\displaystyle u= tan^n^-^2x,,,,,,,,,,,,,,,dv=sec^2xdx$
$\displaystyle du=(n-2)tan^n^-^3xdx,,,,,,v=tanx$

$\displaystyle uv-\int vdu$

$\displaystyle tan^n^-^1x-(n-2)\int tan^n^-^2xdx - \int tan^n^-^2xdx$

so I get
$\displaystyle \int tan^nx = tan^n^-^1x-(n-1)\int tan^n^-^2xdx$ Where's did I go wrong?
• Jul 25th 2010, 02:23 PM
drumist
The derivative of $\displaystyle \tan^{n-2} x$ is not $\displaystyle (n-2) \tan^{n-3} x$. Do you see why?
• Jul 25th 2010, 03:00 PM
Tclack
Oh yeah!
Yeah, the Chain Rule. Thanks for the backup
The text passage contains a mathematical question in the form of a proof for a reduction formula for \(\tan^n(x)\). The author presents their attempt at the proof and asks where they went wrong. An answer is provided by another user, pointing out a mistake in the differentiation.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# tan reduction formula

• Jul 25th 2010, 02:01 PM
Tclack
tan reduction formula
So the proof for the reduction formula for [tan(x)]^n I found here, and it makes sense

But I did this and it is very close, but not the same formula, what's wrong?

$\displaystyle \int tan^nx dx = \int tan^n^-^2x *tan^2x dx$
$\displaystyle =\int tan^n^-^2x(sec^2x -1) dx$
$\displaystyle = \int tan^n^-^2x*sec^2x dx - \int tan^n^-^2x dx$

here I did an integration by parts of the first integral above:

$\displaystyle u= tan^n^-^2x,,,,,,,,,,,,,,,dv=sec^2xdx$
$\displaystyle du=(n-2)tan^n^-^3xdx,,,,,,v=tanx$

$\displaystyle uv-\int vdu$

$\displaystyle tan^n^-^1x-(n-2)\int tan^n^-^2xdx - \int tan^n^-^2xdx$

so I get
$\displaystyle \int tan^nx = tan^n^-^1x-(n-1)\int tan^n^-^2xdx$ Where's did I go wrong?
• Jul 25th 2010, 02:23 PM
drumist
The derivative of $\displaystyle \tan^{n-2} x$ is not $\displaystyle (n-2) \tan^{n-3} x$. Do you see why?
• Jul 25th 2010, 03:00 PM
Tclack
Oh yeah!
Yeah, the Chain Rule. Thanks for the backup
user
The text passage contains a mathematical question in the form of a proof for a reduction formula for \(\tan^n(x)\). The author presents their attempt at the proof and asks where they went wrong. An answer is provided by another user, pointing out a mistake in the differentiation.

TYPE: SINGLE_QUESTION_WITH_ANSWER