Another Way

[Steven G]

Use the Pythagorean Trig Identity, sin^2θ+cos^2θ=1, to prove that sin^4θ-cos^4θ=1-2cos^2θ.

With the advice of my teacher, I started by making a chart. One side is "sin^4θ - cos^4θ", and the other is "1 - 2cos^2θ". He said I'm not supposed to plug in any numerical values, but I'm not sure what to do next.
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In my opinion I think it is usually best to get one side to be what you want as quickly as possible. If you multiply the both sides by sin^2θ-cos^2θ, you'll immediately get the lhs to be sin^4θ-cos^4θ. Then you need to get the rhs, which is now sin^2θ-cos^2θ, to become 1-2cos^2θ. A simple trig identity will get you there (namely a variation of sin^2θ+cos^2θ=1)

 

[mmm4444bot]

In my opinion I think it is usually best to get one side to be what you want as quickly as possible …

… simple trig identity will get you there … [one that's different than the one you were instructed to use] …

I understand your opinion above differs from what appears to be instructor's intent, however, when such opinions do not comport with given instructions, please express them on the Math Odds & Ends board instead of the student's thread.

PS: If you already understood this (i.e., you misread the op or something), please delete post #1 above, and I will remove this thread.

Thank you! :cool:

 

[lookagain]

In my opinion I think it is usually best to get one side to be what you want as quickly as possible.
If you multiply the both sides by sin^2θ-cos^2θ, you'll immediately get the lhs to besin^4θ-cos^4θ. . . .

No, you 1) should be able to show independently how you transform the original expression to the final expression, and
2) show detailed steps to support subsequent steps, as opposed to doing something "as quickly as possible."


The idea of starting on one side and also the other side to meet somewhere in between can be appropriate in a puzzle
such as solving a maze on paper, but that is not analogous to working out a trigonometry identity proof.