# Thread: Struggling to prove Trig Identity

1. ## Struggling to prove Trig Identity

Hi,

I am struggling with trying to prove:

[sin^2(x) +4sin(x) + 3] / cos^2(x) = [3+sin(x) / 1 - sin(x)]

If someone could please walk me through this as well as explaining some tips as to how i can approach similar problems in the future it would be greatly appreciated.
Have tried applying all the identities but I am obviously missing something which is probably obvious...

Thanks

2. ## Re: Struggling to prove Trig Identity

Originally Posted by andy000
Hi,

I am struggling with trying to prove:

[sin^2(x) +4sin(x) + 3] / cos^2(x) = [3+sin(x) / 1 - sin(x)]

If someone could please walk me through this as well as explaining some tips as to how i can approach similar problems in the future it would be greatly appreciated.
Have tried applying all the identities but I am obviously missing something which is probably obvious...

Thanks
I am not sure I can give a general tip on proofs other than trying different ideas out. What do I mean by that?

Let's take this case. I see that what is to be proved is in terms of sine functions and what is given is in terms of sines and cosines, more specifically a cosine squared function. Do I know anything about the relationship about the squares of sines and cosines? I certainly do. So I also can "see" that I can eliminate the squared cosine and get everything in terms of sines and squares of sines. Does that help me? I don't know, but it certainly gives me something to try so

$\dfrac{sin^2(x) + 4sin(x) + 3}{cos^2(x)} = \dfrac{sin^2(x) + 4sin(x) + 3}{1 - sin^2(x)}.$

Here is a trick that is sometimes useful. I have a function that is a built up from a single function, in this case the sine function. It sometimes helps "seeing" how to proceed by simplifying using a substitution.

$Let\ u = sin(x).\ Then\ \dfrac{sin^2(x) + 4sin(x) + 3}{1 - sin^2(x)} = what.$

Can you simplify further now?

3. ## Re: Struggling to prove Trig Identity

Hi Jeff,

I have solved it now by finding the factors and cancelling. Thank you so much for your help, really appreciate you taking the time.

Andy