Can You Simplify this Trig Identity?

[sean trom]

Homework Statement

Show that (sin(2theta) - sin(theta)) / (cos(2theta) - cos(theta) + 1) = tan(theta)

sorry if this setting out is unclear but i am not familiar with how to post math symbols and such.

Homework Equations

above

The Attempt at a Solution

I have tried simplifying using double angle formulae but it seems i am going backwards instead of forwards. This is really frustrating as it seems to be quite simple but when I try to solve, i get nowhere.

Thanks.

 

[Fightfish]

$$\frac{sin (2\theta) - sin (\theta)}{cos (2\theta) - cos (\theta) +1} = \frac{2 sin (\theta) cos (\theta) - sin (\theta)}{2 cos^{2} (\theta) - cos (\theta)} = \frac{sin (\theta)}{cos (\theta)} = tan (\theta)$$

 

[sean trom]

i can see how you got from the first to second stage (using double angle formulae), but how did you come to the 3rd stage, can you please show me smaller steps?

 

[Fightfish]

Factorisation; you will see that the numerator and denominator share a common term, $$2 cos(\theta) - 1$$

 

[sean trom]

Factorisation; you will see that the numerator and denominator share a common term, $$2 cos(\theta) - 1$$

sorry if I am frustrating but i don't quite understand what you mean here. how does the $$2 cos^2(\theta)$$ get eliminated? what happens to the numerator when the $$sin(\theta)$$ gets taken away?

 

[VietDao29]

$$\frac{sin (2\theta) - sin (\theta)}{cos (2\theta) - cos (\theta) +1} = \frac{2 sin (\theta) cos (\theta) - sin (\theta)}{2 cos^{2} (\theta) - cos (\theta)} = \frac{sin (\theta)}{cos (\theta)} = tan (\theta)$$

Fightfish, please do not give full solutions. We are here to guide, and help the OP, not to provide solutions. We can give them a push, or tell them what they should do, or blah blah blah. Giving full solutions is against the forum rules since it does not help much, and isn't beneficial to the OP at all.

Just bear this in mind though:

Give a man a fish, and he will eat for a day. Teach a man to fish, and he will eat for a lifetime. :)

 

[Fightfish]

Sorry, I was under the assumption that the major barrier in such proving questions was usually just a mental block or "lack of inspiration", and hence that it wouldn't hurt to provide the solution in this case

 

[VietDao29]

sorry if I am frustrating but i don't quite understand what you mean here. how does the $$2 cos^2(\theta)$$ get eliminated? what happens to the numerator when the $$sin(\theta)$$ gets taken away?

Factorization means that you take out the common factor, like this:

ab + ac = a(b + c)

Since ab, and ac both have the factor a, so we can "pull" it out.

Another example is:

sin(x) + sin(x)cos(x) = sin(x) (1 + cos(x)), we simply pull sin(x) out.

Now, tell me, how can one factor the numerator, and denominator in your problem?

Sorry, I was under the assumption that the major barrier in such proving questions was usually just a mental block or "lack of inspiration", and hence that it wouldn't hurt to provide the solution in this case

Well, this is one of the fundamental problems, the basic ones. It'd be much better if you can guide the OP through it. He can learn more as he completes the problem on his own. He can do it, it's just that he didn't "see" the pattern. (and this is why he comes here) Or maybe he's lacking some earlier concepts. (we can give him some links, or review them for him) If it's a harder problem, you can give him a push, well.. but I don't think you need to do so in this case. And in some special cases, if it's a very hard problem, and if the OP has put much effort on the problem, but still go no where, then you can post a full solution. But this case is really really rare. :)

 

[sean trom]

yep i understand now, thanks everyone.