Proving a trigonometric identity

In summary, the conversation is about a person needing help with proving a trigonometric identity. They have spent several hours trying different approaches but have not been successful. The conversation also clarifies what a trigonometric identity is and how it differs from a regular equation.
  • #1
egillesp
5
0
Hi,

I need help proving the following trig identity:

\(\displaystyle \frac{\cot^2(x)-\cot(x)+1}{1-2\tan(x)+\tan^2(x)}=\frac{1+\cot^2(x)}{1+\tan^2(x)}\)

Me and my friend have spent several hours determined to figure this out, starting from the left hand side, the right hand side, and doing both together, but nothing seems to work.
For example, I tried to factor the left hand side, \(\displaystyle \frac{(\cot(x)-1)^2}{(\tan(x)-1)^2}\), but it didn't get me anywhere.

Help would be greatly appreciated
 
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  • #2
I tried fixing your $\LaTeX$, and the resulting identity is not true...can you clarify what the actual given identity should be?
 
  • #3
The actual given identity is what I posted in the title and first thing in the thread
 
  • #4
egillesp said:
The actual given identity is what I posted in the title and first thing in the thread

The equation given in your first post is not an identity. :D
 
  • #5
okay, so what do you mean by a trig identity?
 
  • #6
egillesp said:
okay, so what do you mean by a trig identity?

An identity is an equation that is true for all legitimate values of any variables in the equation.

The equation you posted is only true for:

\(\displaystyle x=\frac{\pi}{2}(2k\pm1)\) where \(\displaystyle k\in\mathbb{Z}\)
 
  • #7
okay thank you
 
  • #8
egillesp said:
okay thank you

One of the first things I do when someone posts a trigonometric identity is to use W|A to check to see if it is in fact an identity, because often enough the given identity is in fact not true, and this saves a lot of time and hair pulling. :D

Here is where I checked:

>>click here<<
 

Related to Proving a trigonometric identity

1. What are some basic steps for proving a trigonometric identity?

There are several basic steps that can be followed to prove a trigonometric identity. First, simplify both sides of the equation using known trigonometric identities. Then, use algebraic manipulations to transform one side into the other. Next, use substitution or the Pythagorean identity to simplify further. Finally, check that both sides of the equation are equal.

2. How do I know which trigonometric identities to use when proving an identity?

It is important to have a good understanding of the basic trigonometric identities, such as the Pythagorean identities, double angle formulas, and sum and difference formulas. These can often be used to simplify one side of the equation and make it easier to prove. It may also be helpful to look at the specific trigonometric functions involved and think about which identities might apply to them.

3. What should I do if I get stuck while trying to prove a trigonometric identity?

If you get stuck while trying to prove a trigonometric identity, it can be helpful to step back and review the basic trigonometric identities. You can also try working backwards from the desired result, using algebraic manipulations to see if you can get to the starting point. If all else fails, it can be helpful to consult a textbook or online resource for guidance.

4. Can I use a calculator to prove a trigonometric identity?

No, it is not recommended to use a calculator to prove a trigonometric identity. Using a calculator to solve equations can be helpful, but when proving an identity, it is important to use algebraic manipulations and known identities to simplify both sides of the equation. Relying on a calculator may lead to errors or a lack of understanding of the underlying concepts.

5. Are there any shortcuts or tricks for proving a trigonometric identity?

There are a few common shortcuts or tricks that can be used when proving a trigonometric identity. For example, you can try converting all trigonometric functions to sine and cosine, using the Pythagorean identity to simplify, or looking for patterns in the equations. However, these shortcuts may not always work, so it is important to have a strong understanding of the basic trigonometric identities and be able to use them effectively.

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