trig proof help

[shelly89]

show that \(\displaystyle tanx = x + \frac{1}{3}x^{3} + \frac{2}{15}x^{5} \)

\(\displaystyle tanx = \frac{sinx}{cosx} \)

\(\displaystyle tanx cosx = sinx \)

I know I have to use the power series, so I know the series for cosx and sinx

\(\displaystyle 1-\frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} tanx = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} \)

Not sure what to do from here, could I compare coefficients? how would i write tanx as a polynomial function?

is it

\(\displaystyle a_{0}x_{0} +a_{2}x_{2} + a_{3}x_{3}... \) ??

Any help appreciated thank you

 

[Deleted member 4993]

show that \(\displaystyle tanx = \frac{1}{3}x^{3} + \frac{2}{15}x^{5} \)

\(\displaystyle tanx = \frac{sinx}{cosx} \)

\(\displaystyle tanx cosx = sinx \)

I know I have to use the power series, so I know the series for cosx and sinx

\(\displaystyle 1-\frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} tanx = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} \)

Not sure what to do from here, could I compare coefficients? how would i write tanx as a polynomial function?

is it

\(\displaystyle a_{0}x_{0} +a_{2}x_{2} + a_{3}x_{3}... \) ??

Any help appreciated thank you

Look at your problem and your post carefully!

As posted, the given identity is incorrect.

 

[shelly89]

Look at your problem and your post carefully!

As posted, the given identity is incorrect.

sorry it is a typo, its suppose to be \(\displaystyle tanx = x +\frac{1}{3}x^{3} + \frac{2}{15}x^{5} \) I have corrected my original post

 

[mmm4444bot]

For information, there is a [Preview Post] button next to the [Submit] button. We use it to proofread our typing before we submit our posts. Cheers :cool:

 

[Deleted member 4993]

sorry it is a typo, its suppose to be \(\displaystyle tanx = x +\frac{1}{3}x^{3} + \frac{2}{15}x^{5} \) I have corrected my original post

Now you can divide \(\displaystyle \left (x - \frac{x^3}{3!} + \frac{x^5}{5!} \right) \ by \ \left (1 - \frac{x^2}{2!} + \frac{x^4}{4!} \right)\) using any of the well known methods of polynomial division.

For a quick refresher on polynomial division, go to:

http://search.freefind.com/find.html?id=5014414&pageid=r&mode=ALL&n=0&query=polynomial+division

 

[shelly89]

Now you can divide \(\displaystyle \left (x - \frac{x^3}{3!} + \frac{x^5}{5!} \right) \ by \ \left (1 - \frac{x^2}{2!} + \frac{x^4}{4!} \right)\) using any of the well known methods of polynomial division.

For a quick refresher on polynomial division, go to:

http://search.freefind.com/find.html?id=5014414&pageid=r&mode=ALL&n=0&query=polynomial+division

I know how to do polynomial long division, but I really dont have any idea how to do this one, also my notes say I should form an identity in tanx, and than compare coefficients?

 

[shelly89]

I know tanx is an odd function,

therefore would the polynomial for tanx be \(\displaystyle a_{1}x^{1} + a_{3}x^{3} + a_{5}x^{5} ...... \) ?

so comparing coefficients,

\(\displaystyle x = a_{1}x^{1} , a_{1} = 1 \)

\(\displaystyle \frac{x^{3}}{3!} = a_{3}x^{3} , a_{3} = \frac{1}{3} \)


\(\displaystyle \frac{x^{5}}{12} + \frac{x^{5}}{24} = a_{5}x^{5} \)

for this one I get \(\displaystyle a_{5} = \frac{1}{24} \) when it should be \(\displaystyle \frac{2}{15} \)


any help appreciated.