alexh107
  • alexh107
Verify the trigonometric equation by substituting identities to match the right hand side of the equation to the left hand side of the equation: cot x sec^4 x = cot x + 2 tan x + tan^3 x Please walk me through this step by step. I don't think I've ever been more lost in math.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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mathmale
  • mathmale
Are you sure there are two equal signs in this expression?
alexh107
  • alexh107
Oh sorry, this is the correct one: cot x sec^4 x = cot x + 2 tan x + tan^3 x
mathmale
  • mathmale
that makes a huge difference.

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mathmale
  • mathmale
I see you have no fewer than 3 trig functions in this equation. Can you think of a way to cut that to just 2 trig functions?
alexh107
  • alexh107
Not really. The only thing I know is that you're supposed to start with the right side since it's more complex but after that I don't know what to do.
mathmale
  • mathmale
Need to dig up a table of trig identiies here. Do you have one in front of y ou?
alexh107
  • alexh107
I have the common ones written in my notes.
mathmale
  • mathmale
which one involves (csc x)^2?
alexh107
  • alexh107
1 + cot^2 = csc^2
mathmale
  • mathmale
good. that's correct, accurate.
mathmale
  • mathmale
I see that all terms in your equation involve either tan or cot, except for one: (sec x)^4. Agree or disagree?
alexh107
  • alexh107
Agree
mathmale
  • mathmale
So we might be able to simplify the identity by getting rid of that (sec x)^4. Any ideas on how to do that? How is the sec x function defined?
alexh107
  • alexh107
I'm not really sure. Would we need to do something so that it's like (sec^2)(sec^2) since most of the identities involve sec^2 not sec^4?
mathmale
  • mathmale
That's true. The secant function is defined as 1/ cos x. We don't want to introduce yet another trig function into this inequality, so should not use this. You are correct in that you can re-write (sec x)^4 as [(sec x)^2]^2. Agreed or not?
alexh107
  • alexh107
Agreed
mathmale
  • mathmale
Try using the identity for (sec x)^2 that we discussed earlier.
alexh107
  • alexh107
I got (1 + tan^2) (1 + tan^2) since 1 + tan^2 = sec^2 and we have sec^4 but I'm not sure how to use the one I wrote earlier for 1 + cot^2 = csc^2
mathmale
  • mathmale
Remember, we don't want to introduce or re-introduce more trig functions here. so (csc x)^2 is out. What is the relationship between (tan x)^2 and (sec x)^2?
alexh107
  • alexh107
1 + tan^2 = sec^2
mathmale
  • mathmale
If you can u se Equation Editor, let's use it. But your latest input is clear. enuf (altho you've left out the argument x.). Note that we have (sec x)^4 in that equation. How could we use your latest result to eliminate (sec x)^4 from the equation?
mathmale
  • mathmale
this is Equation Editor output:\[\tan ^{2}x+1=\sec^2x\]
mathmale
  • mathmale
Starting with this latest equation, find an equation for \[\sec^4x\]
alexh107
  • alexh107
\[(\tan^2 x + 1)(\tan^2 x + 1) = \sec^4 x\]
alexh107
  • alexh107
I don't think that's right but I'm not sure what to do.
mathmale
  • mathmale
What you have is fine. this identity allows us to eliminate (sec x)^4 from the equation. Why not multiply out\[(\tan^2x+1)^2\]
mathmale
  • mathmale
and replace (sec x)^4 with the result?
alexh107
  • alexh107
So that would make the first side of the equation: \[\cot x (\tan^2 x +1)^2\]
mathmale
  • mathmale
cot x sec^4 x = cot x + 2 tan x + tan^3 x or\[\cot x \sec^4 x = \cot x + 2 \tan x + \tan^3 x\] Yes, you have the left side correct. Notice that you also have cot x on the right side.
mathmale
  • mathmale
Would it be possible to get rid of the cot x factor?
mathmale
  • mathmale
As you showed me, you now have\[\cot x (\tan^2x + 2 \tan x + 1) \] on the left side, and this is getting to look a lot like the right side. Note that the cot and tan are reciprocals of one another; can you put that fact to good use?
mathmale
  • mathmale
Is this the right side you have also? cot x + 2 tan x + tan^3 x\[\cot x + 2 \tan x + \tan^3 x\]
alexh107
  • alexh107
Yes that is the right side. I'm just confused on the left how we went from \[\cot x (\tan^2 x +1)^2 \to \cot x + 2\tan x +1\]
mathmale
  • mathmale
We've gone thru so much detail already, there's bound to be an oversight somewhere. But note that you can and should expand the square of (tan x)^2 + 1).
mathmale
  • mathmale
I think you've learned a lot here and made some progress towards proving the identity. At this point I ask you which would benefit y ou more, to complete this proof or to move on to another problem.
alexh107
  • alexh107
I can try to move on to another problem I suppose and maybe that one will be easier and I can come back to this one.
mathmale
  • mathmale
I've copied this one down and am gtoing to work on it a bit more. If you'll promise to come back and review what we've done here, then I'd suggest you move on to another proof. Each of us has only so much time, and so it's essential to put every min. to good use.
alexh107
  • alexh107
Okay. I found another one on my paper that looks slightly easier. I can go try to work on that for now.
mathmale
  • mathmale
Hey, I think I've proven the one we were working on. I'll share that with you later. Post the new problem now...Ask a (new) question, not adding on to this already long discussion.
alexh107
  • alexh107
Okay, the new problem I have is:\[1 + \sec^2 x \sin^2 x = \sec^2 x\]
mathmale
  • mathmale
Definitely have finished the proof.
alexh107
  • alexh107
That's great. Thank you for taking the time to help me.
mathmale
  • mathmale
And you're supposed to prove this new identity?
alexh107
  • alexh107
Yes
mathmale
  • mathmale
1 + \sec^2 x \sin^2 x = \sec^2 x\[1 + \sec^2 x \sin^2 x = \sec^2 x\]
mathmale
  • mathmale
the tan and sec functions pair up in an identity, as do the cot and csc.
mathmale
  • mathmale
Also, the reciprocal of the cos is the sec function. Could you use either or both of these facts to begin simplifying the given equation?
mathmale
  • mathmale
Note: the sine and cosine are most familiar to most of us, so y ou might benefit from eliminating the sec function in favor of its reciprocal, the cos.
mathmale
  • mathmale
Simply take the original equation and substitute \[\frac{ 1 }{ \cos^2 x }\]
mathmale
  • mathmale
for (sec x)^2.
mathmale
  • mathmale
just on the left side.
alexh107
  • alexh107
\[1 + 1/\cos^2x + \sin^2x\]
mathmale
  • mathmale
Al, don't we haver multiplication in the left-hand term? You've introduced a 2nd addition.
alexh107
  • alexh107
Oh my bad \[1+1/\cos^2x \sin^2x\]
mathmale
  • mathmale
Basically right; need parentheses around the reciprocal of (cos x)^2. But anyway. What you have now is [1 / (cos x)^2\*[sin x]^2. OK? Can you simplify that?
mathmale
  • mathmale
\[1+\frac{ \sin^2x }{ \cos^2x }=?\]
mathmale
  • mathmale
Leave the 1 alone. The 2nd term is easily simplified.
alexh107
  • alexh107
I'm struggling to find the identity to use. Would it be tan x = sin x/ cos x
mathmale
  • mathmale
yes, preciselyl.
alexh107
  • alexh107
So that means 1 + tan^2 = sec^2?
mathmale
  • mathmale
Yes. is that true or not?
alexh107
  • alexh107
True
alexh107
  • alexh107
So that's the end of that proof?
mathmale
  • mathmale
Then you've proven the identity.
mathmale
  • mathmale
yes.
mathmale
  • mathmale
Now, regarding the previous problem: We have \[\cot x(\tan^4x+2\tan^2x + 1) = \cot x + 2\tan x + \tan^3x.\]
mathmale
  • mathmale
convert that cot x to 1 / tan x and then multiply everything within parentheses by 1 / tan x.
alexh107
  • alexh107
\[2\tan x/\tan x + \tan^3 x /\tan x = \]
mathmale
  • mathmale
Compare that to the right side. See how close you're getting?
mathmale
  • mathmale
Where you should have 2 (tan x)^2, you have written 2 tan x. fix that please.
alexh107
  • alexh107
\[2 (\tan x)^2/\tan x + \tan^3/ \tan x\]
mathmale
  • mathmale
Simplify the left side, then again compare the left and right sides of y our equation.
alexh107
  • alexh107
\[2 \tan x + \tan^2 x\]
mathmale
  • mathmale
Aside from some very minor mistakes you've succeeded in proving this identiy. I encourage you to review our discussion back to what we typed in 8 minutes ago, and make the changes necessary to validate your proof. typing 2 tan x instead of 2 (tan x)^2 was one example. I so much appreciate your involving y ourself in this work as much as you have.
mathmale
  • mathmale
As much as I'd like to continue, I need to shift my focus elsewhere. Hope to have the privilege of working wi th you again soon. :)
alexh107
  • alexh107
I understand. Thank you for your help and spending so much time with me on this. I understood the other proof we completed and hopefully I will be able to figure this one out eventually.
mathmale
  • mathmale
Cool. It's worth the effort! See you, Alex! Bye.

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