Prove trigonometric identity and determine a counterexample

In summary, the given equation is an identity because it can be simplified to cos(x-y)cosy - sin(x-y)siny, which is equivalent to cosx. Using the trigonometric identity cos(A+B) = cosAcosB - sinAsinB, we can rewrite the equation as cos(x-y+y) = cosx, which simplifies to cosx = cosx, proving that the given equation is an identity.
  • #1
euro94
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0

Homework Statement


cos(x-y)cosy-sin(x-y)siny=cosx
a.try to prove that the equation is an identity
b. determine a counterexample to show that it is not an identity

Homework Equations


cos(x-y) = cosxcosy+sinxsiny
sin(x-y) = sinxcosy-cosxsiny


The Attempt at a Solution


a.Left side of equatioin: (cosxcosy+sinxsiny)cosy - (sinxcosy-cosxsiny)siny
= cosxcosycos2y+sinxcosysinycosy - (sinxcosysiny - cosxsin2y)
I'm not sure where to go from there ...
b. how would i go about finding a counterexample?
 
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  • #2
(cosxcosy+sinxsiny)cosy - (sinxcosy-cosxsiny)siny
= cosxcosycos2y +sinxcosysinycosy - (sinxcosysiny - cosxsin2y)

That part is wrong. Once you get that part right, consider trying this out: Factor cos(x) from two of the clusters of terms above and a simplification will happen.For the counterexample, just find a value of x and a valuye for y so that the equality doesn't hold.
 
Last edited:
  • #3
euro94 said:

Homework Statement


cos(x-y)cosy-sin(x-y)siny=cosx
a.try to prove that the equation is an identity
b. determine a counterexample to show that it is not an identity

Homework Equations


cos(x-y) = cosxcosy+sinxsiny
sin(x-y) = sinxcosy-cosxsiny

The Attempt at a Solution


a.Left side of equatioin: (cosxcosy+sinxsiny)cosy - (sinxcosy-cosxsiny)siny
= cosxcosycos2y+sinxcosysinycosy - (sinxcosysiny - cosxsin2y)
I'm not sure where to go from there ...
b. how would i go about finding a counterexample?

Parts a and b are mutually exclusive. Either the relation given is an identity, or it is not. If it's an identity, you're supposed to prove it as in part a (in which case you don't have to answer part b). If it's not an identity, you can just provide a single counterexample for part b (in this case, you can't answer part a).

For this question, it is, in fact an identity. So only part a has an answer.

You know that cos(A+B) = cosAcosB - sinAsinB.

Now try letting A = x-y and B = y. What happens?
 

1. What does it mean to "prove a trigonometric identity"?

Proving a trigonometric identity involves using mathematical techniques to show that an equation or statement involving trigonometric functions is true for all values of the variables involved. This often involves using algebraic manipulations and trigonometric identities to simplify both sides of the equation until they are equivalent.

2. Why is it important to prove trigonometric identities?

Proving trigonometric identities is important because it allows us to verify the validity of equations and statements involving trigonometric functions. This can be useful in solving more complex mathematical problems and in real-world applications such as engineering and physics.

3. What is a counterexample and how does it relate to proving trigonometric identities?

A counterexample is a specific example or case that disproves a statement or equation. In the context of proving trigonometric identities, a counterexample can be used to show that an identity is not always true, even if it appears to be true for some values. This helps to ensure that the identity is valid for all possible values of the variables involved.

4. Can you provide an example of proving a trigonometric identity?

Sure, an example of proving a trigonometric identity is showing that sin^2(x) + cos^2(x) = 1 for all values of x. This can be done by using the Pythagorean identity, sin^2(x) + cos^2(x) = 1, and substituting in any value for x to demonstrate that the equation holds true.

5. Are there any tips for successfully proving trigonometric identities?

Yes, some tips for successfully proving trigonometric identities include familiarizing yourself with common trigonometric identities, carefully analyzing the given equation or statement, and using algebraic manipulation and substitution to simplify both sides of the equation until they match. It can also be helpful to break down the problem into smaller steps and to practice with different types of identities to build understanding and confidence.

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