1. Proving an identity

Hi,

I hope someone can help. I'm trying to understand the process of proving an identity.

Let's say that you were trying to prove the following equation: sin2x/(1+cos2x) = tanx

I know that the equation is an identity since when I simplify the left-hand side, it is equivalent to the right-hand side.

But let's say that I replaced x with pi/4, and still showed that both sides were equivalent - would using only one value be enough to prove that the equation is an identity? I would think that by proving the equation using only one value (e.g. pi/4), would not be enough to prove that the equation is an identity. I thought that x takes into account all values, so it is more accurate than merely subbing in pi/4.

The definition of a trigonometric identity is as follows:

I want to place emphasis on the fact that the definition says that the solution set for trigonometric identities is all real numbers (and not including the solutions that are undefined). So with that being said, if x was only equal to pi/4, then how could the equation possibly be an identity?

I would appreciate some guidance.

Sincerely,
Olivia

2. Re: Proving an identity

Originally Posted by otownsend
Let's say that you were trying to prove the following equation: sin2x/(1+cos2x) = tanx

I know that the equation is an identity since when I simplify the left-hand side, it is equivalent to the right-hand side.
But let's say that I replaced x with pi/4, and still showed that both sides were equivalent - would using only one value be enough to prove that the equation is an identity? I would think that by proving the equation using only one value (e.g. pi/4), would not be enough to prove that the equation is an identity. I thought that x takes into account all values, so it is more accurate than merely subbing in pi/4.

The definition of a trigonometric identity is as follows:

I want to place emphasis on the fact that the definition says that the solution set for trigonometric identities is all real numbers (and not including the solutions that are undefined). So with that being said, if x was only equal to pi/4, then how could the equation possibly be an identity?
Note that $2\left( {\dfrac{\pi }{4}} \right) = \left( {\dfrac{\pi }{2}} \right)$ so

\begin{align*}\dfrac{\sin \left( {\frac{\pi }{2}} \right)}{1-\cos \left( {\frac{\pi }{2}} \right)}&=\dfrac{1}{1-0}\\&=1\\&=\tan \left( {\frac{\pi }{4}} \right) \end{align*}

3. Re: Proving an identity

They gave an example of it working. It may be left up to the reader to prove the identity. Such as this:

$\dfrac{\sin(2x)}{1+\cos(2x)} = \dfrac{2\sin x\cos x}{1+2\cos^2 x-1} = \dfrac{\sin x \cancel{2\cos x}}{\cancel{2\cos x} \cos x} = \dfrac{\sin x}{\cos x} = \tan x$

This proves the identity.

4. Re: Proving an identity

Thank you for responding... however, I'm more wondering whether it is incorrect style to prove an identity by solving for a specific variable as opposed to just x?

5. Re: Proving an identity

Originally Posted by otownsend
Thank you for responding... however, I'm more wondering whether it is incorrect style to prove an identity by solving for a specific variable as opposed to just x?
A statement, an identity, is never proved by example.
An example can disprove a statement called a counter-example; but it can never ever prove a statement.

6. Re: Proving an identity

What if the identity held true to an example such as when x = pi/4? What would that be communicating? Based on what you have told me, I would think that this would not be enough to claim the whole equation for all cases as an identity. Please let me know if my thinking about this is right...

7. Re: Proving an identity

Originally Posted by otownsend
What if the identity held true to an example such as when x = pi/4? What would that be communicating? Based on what you have told me, I would think that this would not be enough to claim the whole equation for all cases as an identity. Please let me know if my thinking about this is right...
No, it just proves that the world is flat.

8. Re: Proving an identity

Right, and it also proves that global warming isn't real.

9. Re: Proving an identity

Originally Posted by otownsend
Right, and it also proves that global warming isn't real.