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

[RikB]

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?

 

[Bohrok]

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.

 

[RikB]

I have to do it in a formal math proof format so I can't add 1 on another side.

 

[Mark44]

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

 

[Bohrok]

Hmm, "adding one" must be something between adding zero and multiplying by one. Corrected.

 

[RikB]

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.

 

[Bohrok]

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.

 

[Mark44]

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.

 

[Robertdavids]

Ive been looking for a refresh on Trig thanks a million

 

[arildno]

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.

 

[zgozvrm]

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.