Helpjebal
  • Helpjebal
cot x sec4x = cot x + 2 tan x + tan3x I have to make the left side look like the right to prove this equation is true. I've done most of it but I'm stuck at the end cot x sec^4x = cot x + 2 tan x + tan^3x cot x sec^2x sec^2x = cot x + 2 tan x + tan^3x cot x + 1 + tan^2 x + 1 + tan^2 x = cot x + 2 tan x + tan^3x
Mathematics
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SOLVED
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jamiebookeater
  • jamiebookeater
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Helpjebal
  • Helpjebal
cot x + 2 + tan^2 x +tan^2 x
Helpjebal
  • Helpjebal
sorry the right side has to look like the left side
Helpjebal
  • Helpjebal
now i'm really confused....

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anonymous
  • anonymous
this might help you: cos2x = 2cos^2(x) -1
anonymous
  • anonymous
it doesnt matter which side should look like which one
anonymous
  • anonymous
as long as you can prove that you can derive one side from the other, your proof is correct
mathmale
  • mathmale
Yes, and you might also think about relevant trig identities: cot x = (cos x) / (sin x) sec x = 1 / (cos x) These are just examples. You might also try tan x = (sin x) / (cos x)
anonymous
  • anonymous
i'd say start with the right side. it'll be easier
anonymous
  • anonymous
left*
Helpjebal
  • Helpjebal
can I just do this? cot x + 2 + tan x + tan^3 x cot x + 2 + tan^4 x cot x + 2(1 + tan^2 x) cot x + 2(sec^2 x) cot x +sec ^4 x which would just be cot x sec ^4 x like in the original?
anonymous
  • anonymous
is it sec^4x or sec4x in the left side?
mathmale
  • mathmale
@Helpjebal: your proposed expression shows ' 2 ' standing by itself. This is not the case in the given, original, equation. So, unfortunately, my answer is "no."
Helpjebal
  • Helpjebal
it's sec^4x on the left side
anonymous
  • anonymous
and how did you get tan^3x from tan3x?
anonymous
  • anonymous
oh
mathmale
  • mathmale
You might have to experiment (e. g., start with the left side, start with the right side) until you find yourself on the right track, especially if you're not used to this sort of problem. Again I urge you to consider the trig identities I typed to you. So far I haven't seen you using them.
Helpjebal
  • Helpjebal
I didn't write it properly, it's cot x sec^4x = cot x + 2 tan x + tan^3x
mathmale
  • mathmale
I agree that you can work on either side, or on both sides. If you change the right side, you can always change it back to its original form in the end. The key to proving this identity lies in the proper use fo trig identities. What is the common trig identity that shows the relationship between the cot and the sec functions?
mathmale
  • mathmale
Hint: this identity stems from the even more common\[\sin^2x+\cos^2x=1\]
Helpjebal
  • Helpjebal
okay so I think I know how to use this trig identities you showed earlier... cot x + 2 tan x + tan^3 x 1/ tan x + 2 tan x + tan^3 x (1 / tan x) + (2 tan x + tan^3 x / 1) 1 + 2 tan x + tan^3 x / 1 tan x that's where I've gotten... but you said that sin^2x+cos^2x=1 so I could also do cot x + 2 tan x + tan^3 x cos(x)/sin(x) + 2(sin(x)/cos(x)) + tan^3x?
mathmale
  • mathmale
The trick here is to reduce the number of different trig functions in your expressions. Of course you can define tan x as (sin x) / (cos x), but do you really want to introduce sin x and cos x here, when the primary trig functions of the given expression are cot x and sec x? I think not.
mathmale
  • mathmale
Have y ou a table of trig identities available? If so, review \[\sin^2x + \cos^2x = 1\]
mathmale
  • mathmale
and find the identity involving \[\cot^2x\]
mathmale
  • mathmale
Then apply this "new" identity to simplifying the given expression. Again: seek to REDUCE (not increase) the number of trig functions you're using.
anonymous
  • anonymous
ok i am back
anonymous
  • anonymous
i did the question. there are only three steps to solving it
anonymous
  • anonymous
first, you want to convert the sec^4x to something with a lower order
anonymous
  • anonymous
use the identity sec^x = 1+tan^2x
anonymous
  • anonymous
sec^2x = 1+tan^2x
Helpjebal
  • Helpjebal
Thats how I was originally trying to solve it cot x + 1 + tan^2x + 1 + tan^2x cot x + 2 + tan^4 x but I got incredibly stuck...
Helpjebal
  • Helpjebal
The only identity i know of involving cot^2 x is + cot^2x = csc^2x
mathmale
  • mathmale
Please double check before you make substitutions such as "cos x + 1." That is NOT an identity.
Helpjebal
  • Helpjebal
I never said cos x + 1... 2(1 + tan x) is the same as 1 + tan^2x + 1 + tan^2x thats why i wrote it like that
mathmale
  • mathmale
Check out physo's suggestion: sec^x = 1+tan^2x. Actually, this should be (sec x)^2 = 1 + (tan x)^2. Is it correct? Double check. If yes, apply it to simplifying the given expression.
anonymous
  • anonymous
i'll give you the first step to help you:\[cotx(\sec^2x)^2\]
mathmale
  • mathmale
@Helpjebal : You did type in the following: cot x + 1 + tan^2x + 1 + tan^2x cot x + 2 + tan^4 x
anonymous
  • anonymous
now use the identity on sec^x and expand it
anonymous
  • anonymous
sec^2x
mathmale
  • mathmale
Let's let physo help here. I'll be in the background if you need me.
Helpjebal
  • Helpjebal
okay so physo we're on the left side right? so... cot x sec^4x cot x (sec^2 x)^2 cot x (1 + tan^2x)^2 am I on the right path?
anonymous
  • anonymous
yes
anonymous
  • anonymous
do you remember (a+b)^2 = a^2 + 2ab + b^2?
Helpjebal
  • Helpjebal
yes I remember that so... cot x sec^4x cot x (sec^2 x)^2 cot x (1 + tan^2x)^2 cot x + 1 + 2 tan^2 x + tan^4x
anonymous
  • anonymous
you need to take the brackets into consideration
mathmale
  • mathmale
this is fine: cot x sec^4x cot x (sec^2 x)^2 cot x (1 + tan^2x)^2 but the next line is not.
anonymous
  • anonymous
yup
mathmale
  • mathmale
physo is right: those parentheses / brackets are very important gatekeepers. We can't just ignore them. Anything within ( ) must be done first, followed by any exponentiation.
Helpjebal
  • Helpjebal
okay so (1 + tan^2x)(1 + tan^2x) 1 + tan^2x + tan^2x + 2 tan^2x 1 + tan^4x + 2 tan^2x making that correction it would be cot x sec^4x cot x (sec^2 x)^2 cot x (1 + tan^2x)^2 cot x + 1 + tan^4x + 2 tan^2x
Helpjebal
  • Helpjebal
wait sorry
Helpjebal
  • Helpjebal
i did that wrong again didn't i...
mathmale
  • mathmale
Please note: because that cot x is OUTSIDE your parentheses, you must multiply everything INSIDE the parentheses by cot x. Distributive rule of multiplication. ;(
anonymous
  • anonymous
yes mathmale is correct. you squared it correctly but you forgot to multiply it with cotx
Helpjebal
  • Helpjebal
ohhhhhh cot x (1 + tan^4x + 2 tan^2x) cot x + tan^3 + 2 tan x
anonymous
  • anonymous
there you go
mathmale
  • mathmale
Hold it, physo. That wasn't my work.
Helpjebal
  • Helpjebal
Thank you guys so much! I'm so incredibly thankful!
mathmale
  • mathmale
So glad you've been able to solve this problem!

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