Understanding the Expansion of Large Tan(x)

[physlad]

Hi, I was reading a review and I saw this equation,

$$A^2 = \frac{|B^2 - C^2|}{\sqrt{(1-sin^2(2x))}} - C^2 - B^2 - 2D^2$$

Then at some point he writes: "expanding for large $tan(x)$, this expression becomes,

$$A^2 = -2(C^2 + D^2) + \frac{2}{tan^2(x)}(B^2 - C^2) + O(1/tan^4(x))$$

Could anybody explain how did this happen?

 

[JJacquelin]

Let t = 1/tan(x)
Compute sin(2x) as a function of t.
Bring it back into the equation.
Develop it as a a series for t.
When tan(x) tends to infinity t tends to 0.
Remplace t by 1/tan(x) in the series.
You will obtain the expected result.

 

[physlad]

Thank you very much, JJacquelin! now I see it :D