I've asked about this same question before, and someone gave me the way to finish, which I understand to some extent. I need help figuring out what they did in the second step though. How they got to the third step from the second.

[sinx + tanx]/[cosx + 1] = tanx

LS=[sinx+tanx]/[cosx+1]
=[sinx+tanx]/[cox+(cosx/cosx)]

makes sense so far, but I don't know how they got from here to the next one

=[sinxcosx-sinx+sinx-sinx/cosx]/[cosx-1]

or to the next

=[(sinxcos^2x-sinx)/cosx]/[cos^2x-1]

but from there I understand

=[sinx(cos^2x-1)/cosx]*[1/cos^2x-1]
=sinx/cosx
=tanx
=RS

Help?

It looks like you cut-and-pasted part of my solution I gave you for this question

http://www.jiskha.com/display.cgi?id=1276915220

I multiplied top and bottom by (cosx -1)/cosx -1) , which of course has a value of 1 and does not change the value of the Left Side.

It is a step similar to the one we use in "rationalizing a denominator", I noticed that this would yield cos^2x -1 , which then could be replaced by the single term sin^2x

Wow that was a tough one... It took me so long to figure out, even with your help! Thanks Reiny! :)

Prov that 1-(sinxtanx) / 1+sec x

To understand how the given expression simplifies from the second step to the third step, let's break it down step by step:

Starting from the second step:
LS = (sinx + tanx) / (cosx + 1)
We want to simplify the denominator by creating a common denominator with cosx.

Note that 1 can be written as cosx/cosx. So the expression becomes:
LS = (sinx + tanx) / (cosx + (cosx / cosx))

Now, consolidate the denominator using the same denominator for the two terms:
LS = (sinx + tanx) / [(cosx * cosx + cosx) / cosx]

Next, simplify the denominator:
LS = (sinx + tanx) / [(cosx * (cosx + 1)) / cosx]

Now, let's simplify the numerator:
LS = [sinxcosx + sinxtanx] / [(cosx * (cosx + 1)) / cosx]

Now, simplify further by distributing cosx to both terms in the numerator:
LS = [sinxcos^2x + sinxtanx] / [(cosx * (cosx + 1)) / cosx]

Combine the terms in the numerator:
LS = (sinxcos^2x - sinx + sinx - sinx) / [(cosx * (cosx + 1)) / cosx]

Simplify the numerator by combining like terms:
LS = (sinxcos^2x - sinx) / [(cosx * (cosx + 1)) / cosx]

Now, let's simplify the denominator:
LS = (sinxcos^2x - sinx) / [(cos^2x + cosx) / cosx]

Finally, rewrite the denominator as (cos^2x - 1):
LS = (sinxcos^2x - sinx) / [(cos^2x - 1) / cosx]

Therefore, the expression simplifies to:
LS = (sinxcos^2x - sinx) / (cos^2x - 1)

From there, you can continue simplifying the expression, as you mentioned in your question, to reach the RS.