Proving cos^4(x)= 3/8+1/2(cos2x)+1/8(cos4x)

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Homework Help Overview

The discussion revolves around proving the identity cos4(x) = 3/8 + 1/2(cos(2x)) + 1/8(cos(4x)). The subject area is trigonometric identities and transformations.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss transforming cos2(x) using the identity cos2(x) = (1 + cos(2x))/2 and squaring both sides to derive cos4(x). There are questions about manipulating terms and ensuring correct substitutions, particularly regarding the handling of cos(2x) and cos(4x).

Discussion Status

Participants are actively exploring different approaches and clarifying steps in their reasoning. Some have identified mistakes in their arithmetic and are encouraged to proceed carefully. There is no explicit consensus yet, but guidance has been offered regarding the use of trigonometric identities.

Contextual Notes

There are indications of confusion regarding the transformations and substitutions, particularly with the variables involved in the trigonometric functions. Participants are also navigating the constraints of homework rules that may limit their ability to seek direct solutions.

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Homework Statement



prove that cos^4(x)= 3/8 +1/2(cos2x)+1/8(cos4x)

Homework Equations





The Attempt at a Solution



i attempted to change the cos2x to 2cos^2(x)-1...then made all them over 8...and now I am stuck...another question...if i multiply 2cos(4x) by 2, do i multiply the 2 and the 4x? or just he 4x?
 
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Well...Use the fact that cos^2(x) = (1 + cos(2x))/2 ...Then square both sides...you will then be pretty close...(that is..you will have cos^4(x) = some mess)..you will just need to use that above formula again to reduce the (cos^2(2x)) you get after squaring the right hand side of the equation...make sure you realize that when plugging (2x) into the left hand side...you will need to make the same plug in in the right...after that...the problem should unfold nicely... You just needed a different formula..
 
do i isolate the cos(2x) in cos^2(x) = (1 + cos(2x))/2 and sun that in?
 
.. I don't know what you mean.. With this method...that is starting with the formula "cos^2(x) = (1 + cos(2x))/2", after squaring and showing that the left hand side is indeed Cos^4(x), your goal is to git rid of every square you have on the right hand side of that...and it turns out, there is only 1...so you use the formula again, but make sure you keep track of that doubled variable...
 
quick...and dumb question...

i get 3+cos2x+2cos2x all over 8...which id close except the cos2x should be a cos4x
 
Ha! Try again...Your are making mistakes in your arithmetic.. Just take it nice and slow... The above isn't as close as you might think...because not only should a cos2x be a 4x...but 2cos2x divided by 8...isnt 1/2cos2x...
 
lmao...il try it again slow
 
ha...tell me what you get
 
ahhhh i got it lol...when i was working out the cos^2(2x), i wrote it originally as 1+cos2x over 2, instead of 1+cos4x over 2...is that right now?
 
  • #10
I told you to watch out for that 2x! ><
 

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