Expert: Sherry Wallin - 12/16/2008

Question
Can you show me the step by step process for answering this question.
((sin)/(1+sin))-((sin)/(1-sin))

thanks


Answer
Hi Jake~
    The way to simplify the is problem is to find the common denominator. If you wanted to add 3/8 + 1/3 you would decide that the common denominator was 24 correct? And how would you come to that conclusion? Do you remember from algebra conjugates? sum and difference of two terms? well 1 + sin x and 1 - sinx are conjugates as well. By the way sin means nothing without the argument (x, theta, degrees etc). Also I assume you were asked to simplify, there really is not a question here to answer. But based on the fact that I think you are to simplify and you meant to put in x for the angle here is the way to do this problem:

sin/(1+sinx)-sinx/(1-sinx)  now multiply top and bottom of the first term by (1-sinx) and multiply top and bottom of the 2nd term by (1+sinx).
This is a legal maneuver because all we've done is multiplied each of the terms by 1.
(1-sinx)sinx    -    sinx(1+sinx)
------------          -------------  
(1-sinx)(1+sinx)   (1+sinx)(1-sinx)

= (sinx - sin^2x -sinx -sin^2x)/(1-sin^2x)
= -2sin^2x/cos^2x
= -2 tan^2x since tan x = sinx/cosx

Hopefully this makes sense, if you still have questions send me a follow up question. Good luck,

Math Prof