Proof Sin^2(x)-Sin^2(2x)=Cos^2(2x)-Cos^2(x) - Get Help Now!

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Discussion Overview

The discussion revolves around proving the trigonometric identity sin^2(x) - sin^2(2x) = cos^2(2x) - cos^2(x). Participants are exploring various methods and formal proof requirements related to this identity, with a focus on the appropriate mathematical techniques to use.

Discussion Character

  • Debate/contested
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant requests assistance in proving the identity and expresses confusion over previous attempts.
  • Another suggests adding zero to the left side and using the identity sin^2(x) + cos^2(x) = 1 to simplify the expression.
  • A participant insists on the necessity of adhering to formal proof formats, stating that one cannot add a number to both sides of the equation.
  • In response, another participant argues that adding a number to both sides is valid under the addition property of equality.
  • One participant humorously acknowledges a misunderstanding regarding the concept of "adding one."
  • Another participant shares their instructor's advice to start from one side of the equation and not alter the other side during formal proofs.
  • One suggestion involves manipulating the left side by adding and subtracting cos(2x) to align it more closely with the right side.
  • Several participants reiterate the importance of starting from one side in formal proofs while discussing the validity of operations applied to both sides of an equation.
  • One participant expresses gratitude for a refresh on trigonometry, indicating a desire to improve their understanding.
  • Another participant cites the fundamental identities cos^2(x) + sin^2(x) = 1 and cos^2(2x) + sin^2(2x) = 1 to support their argument.
  • One participant emphasizes that applying reversible operations to both sides does not change the truth of the equation, while cautioning against non-reversible operations.

Areas of Agreement / Disagreement

Participants exhibit disagreement regarding the methods of proof, particularly around the validity of adding terms to both sides of the equation. Some advocate for starting from one side only, while others argue that manipulating both sides is acceptable as long as the operations are reversible. The discussion remains unresolved with multiple competing views on the approach to proving the identity.

Contextual Notes

Participants reference formal proof requirements and the validity of mathematical operations, indicating a focus on the structure of proofs rather than the specific trigonometric identity itself. There are unresolved assumptions about the definitions and applications of the identities discussed.

RikB
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sin^2(x)-sin^2(2x)=cos^2(2x)-cos^2(x)

I need help with proving this trig identity. Every thing I've tried just makes the problem more confusing. How would you guys go about this?
 
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You can try "adding zero" (not 1, sorry) to the left side and make use of sin2x + cos2x = 1 to get rid of the terms with sine and be left with cosine terms to equal the right side.
 
Last edited:
I have to do it in a formal math proof format so I can't add 1 on another side.
 
There is nothing informal or incorrect about adding a number to both sides of an equation. The addition property of equality can be invoked to do this.

If a = b then a + c = b + c
 
Hmm, "adding one" must be something between adding zero and multiplying by one. :blushing: Corrected.
 
My past instructor said that in a formal proof always start from one side to get to the other and not to alter the other side.
I'm sure if you can solve it by adding a one on the other side of the equation you can also solve it without.
 
Yes, you can do it by only manipulating one side. What I had in mind was adding and subtracting cos2x on the left side and using an identity to make the left side look more like the right side. Then you can do the same thing for the other term.
 
RikB said:
My past instructor said that in a formal proof always start from one side to get to the other and not to alter the other side.
I'm sure if you can solve it by adding a one on the other side of the equation you can also solve it without.

It is valid to apply an operation to both sides of an equation as long as the operation you apply to both sides is reversible, such as adding 1, or multiplying by some nonzero constant. As long as you perform operations such as these, the solution sets of the two equations are identical. What is not generally valid is applying a nonreversible operation like squaring. With an operation such as this, the two equations are not guaranteed to have the same solution sets.
 
Ive been looking for a refresh on Trig thanks a million
 
  • #10
RikB said:
My past instructor said that in a formal proof always start from one side to get to the other and not to alter the other side.
I'm sure if you can solve it by adding a one on the other side of the equation you can also solve it without.

We have:
\cos^2(x)+\sin^{2}(x)=1
as well as:
\cos^2(2x)+\sin^{2}(2x)=1

Thus, the left-hand sides must equal each other.
 
  • #11
RikB said:
My past instructor said that in a formal proof always start from one side to get to the other and not to alter the other side.

In trying to prove that an equation is true, there is nothing wrong (or "informal") about starting with that equation (and assuming that it is true) and then applying a mathematical operation to both sides, because, by doing so, you haven't changed the statement.

So, if you can get the equation reduced to something more familiar that you know is true (say, "X = X" or "1 = 1"), then you have proven that the original equation is true.

Besides, if this isn't for homework, then there is no instructor limiting how you go about your proof. As long as you stick to the rules of mathematics, you'll be okay.
 

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