prove that each of the following identities is true see below

Question

anonymous · Answer

$$\cos^2x(1+\tan^2x)=1$$

anonymous · Answer

@robtobey  or @whpalmer4

kirbykirby · Answer

There is the identity: $1+\tan^2x=\sec^2x$

And by definition: $\large \sec x = \frac{1}{\cos x}$

kirbykirby · Answer

If you square the last equation above on both sides, I hope you'll easily see how you get 1 :)

e.mccormick · Answer

On proofs, you work on only one side of the = and until it looks like the other.

Can you think of another thing that is = to 1 in trig?

anonymous · Answer

well $$\sin^2\theta+\cos^2\theta=1$$

e.mccormick · Answer

So, think you can get it into that form?

anonymous · Answer

??

anonymous · Answer

distribute the cos^2 (x)

e.mccormick · Answer

There are a few tricks to doing identities... well, tips that can help.  One is putting everything in sine and cosine.  No tangents.  This can help you see things that can combined or cancel.

When you do that, it may help you see how to get it into that other form that is 1.

anonymous · Answer

so $$\cos^2x(1+\frac{ \sin(x) }{ \cos(x)}$$

e.mccormick · Answer

Yep.  Now, what would you do next?

anonymous · Answer

would u have to distribute ?

e.mccormick · Answer

Yep!

anonymous · Answer

$$\cos^2x+\frac{ \sin x(\cos^2x) }{ \cos(x) }$$
??

e.mccormick · Answer

You missed the $^2$ on the bottom, but otherwise it is fine.

e.mccormick · Answer

Ah, I see, you made the mistake a little earlier too.  The tan was squared, so that entire fraction is squared.

e.mccormick · Answer

$tan^2x=\dfrac{\sin^2x}{\cos^2x}$

anonymous · Answer

so when i wrote it out should of been 
$$\cos^2x(1+\frac{ \sin^2x }{ \cos^2x })$$

e.mccormick · Answer

Yes.

e.mccormick · Answer

Do you see how it works from there?  And this is only one way to do it.

anonymous · Answer

yes i distribute still right

e.mccormick · Answer

Yep.

anonymous · Answer

$$\cos^2x+\frac{ \sin^2x(\cos^2x) }{ \cos^2x}$$
and cancel on fraction right :)

e.mccormick · Answer

Yep.  Then what do you have?

anonymous · Answer

$$\cos^2x+\sin^2x =1$$

e.mccormick · Answer

And you can replace the left side of that with?

e.mccormick · Answer

See, the goal is to get the left and right of the = to exactly match.  Otherwise it is not proven.

anonymous · Answer

:)

anonymous · Answer

so its done ? right

e.mccormick · Answer

Well, once you have 1 = 1, it is done.  Not written formally, but done.

anonymous · Answer

:) ok thanks :)

e.mccormick · Answer

Proofs are a formal process.  It is important to use the formality because they are used to prove what you know.

Here are some basic rules:

You use the implies arrow, $\implies$, to show the relationship.  Then you use therefore, $\therefore$ to state what it gets you.  And at the end you put Q.E.D. or // to say $\textit{quod erat demonstrandum}$ or "That which has been demonstrated."


$\cos^2x(1+\tan^2x)=1$

$\implies \cos^2x\left(1+\dfrac{\sin^2x}{\cos^2x}\right)=1$

$\implies \cos^2x+\cos^2x\cdot\dfrac{\sin^2x}{\cos^2x}=1$

$\implies \cos^2x+\sin^2x=1$

$\implies 1=1$

$\therefore \cos^2x(1+\tan^2x)=1$

//


OR:

You can work it the way @kirbykirby was talking about.  It is just as good.  He was going to use $1+\tan^2x=\sec^2x$ and $\sec^2 x = \dfrac{1}{\cos^2 x}$ in a similar way.

$\cos^2x(1+\tan^2x)=1$

$\implies \cos^2x(\sec^2x)=1$

$\implies \cos^2x\left(\dfrac{1}{\cos^2 x}\right)=1$

$\implies 1=1$

$\therefore \cos^2x(1+\tan^2x)=1$

//

If you see what he saw, it can go more rapidly.  But until you really know the assorted identities really well, it can be effective to start by putting everything in sine and cosine, then working from there.

kirbykirby · Answer

Yup. Many ways to do proofs :)

A good thing maybe when you start, is to have a table of trig identities in front of you so you can work with them. As you do more proofs, you'll start to identify which identities might be more useful in some situations more than others.

anonymous · Answer

thanks so much so helpful :)

e.mccormick · Answer

Yah, like one of the trig sheets from here: http://tutorial.math.lamar.edu/cheat_table.aspx

e.mccormick · Answer

This video set also does a lot of them. In the homework sections he works problems. http://www.youtube.com/watch?v=QztUOagckfs&list=PLC849CCC8B64352C8 The GCCC lectures are a great way to get a second look at a professor's trig lecture.

anonymous · Answer

thanks will add that to my list - i like to try figure things out before class so havent actually done this yet in class !