Proving Identities Trig final review

[Louise Johnson]

Finally starting to understand using identities!! However I have a problem with one question in particular. I think it is most likely some simple identity that I am overlooking or simply don't have in my text. You can see that both sides are equal however are recprical of each other. Anyways I tried using identities for the 1's on either side and swapping out cosx and sinx.
Please help.

Thank you

Louise

cosx/1-sinx =1+sinx/cosx

 

[stapel]

cosx/1-sinx =1+sinx/cosx

What you have posted means:

. . . . .cos(x) - sin(x) = 1 + tan(x)

...since cos(x) / 1 = cos(x) and sin(x)/cos(x) = tan(x). Is this what you meant? Or did you mean something such as the following?

. . . . .cos(x) / [1 - sin(x)] = [1 + sin(x)] / cos(x)

Thank you.

Eliz.

 

[Louise Johnson]

Sorry I always have difficulty using this calculator notation. The below is what I meant.
Thank you
Louise

cos(x) / [1 - sin(x)] = [1 + sin(x)] / cos(x)

 

[stapel]

The "obvious" solution would be to cross-multiply, but that isn't a valid "prove this identity" solution method, since you have to work with one side only, to get it to convert to the other side. But that does provide a very strong hint of how to proceed.

Since you would like to get "cos²(x) = 1 - sin²(x)", something you know to be true, but since you cannot do the cross-multiplication, "fake it" instead.

Hint: What happens if you multiply the left-hand side by [1 + sin(x)] / [1 + sin(x)]?

Eliz.

 

[Louise Johnson]

Ahh you see I wasn't aware you could do such a thing. Solved it no problem.
Thank you
Louise

 

[stapel]

This is a sort of "working backwards" of the sort that you rarely see actually pointed out, let alone explained, in textbooks. But think about some proof or explanation you've seen, where you wondered "How the Sam hill did they know to do that?!?", and that's probably a place where the writer worked backwards to figure out how to work it, and then just showed you the magically perfect solution.

This is why God invented scratch paper. :wink: Don't be afraid to fiddle with the equations, maybe doing things that aren't, strictly speaking, "correct", if it helps you get the feel of what's going on and what you need to do.

Eliz.