Verify that each equation is an identity

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In summary, the question is asking to verify if each equation is an identity. The conversation discusses using the double-angle formula to factorize and simplify the equation, but there is some confusion about whether the factorization is necessary. The final conclusion is to not lose track of the right side of the equation.
  • #1
kuahji
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The question is "Verify that each equation is an identity."

Problem
tan 8k - tan 8k tan^2 4k = 2 tan 4k

The first thing I tried was to factor out the tangents.

tan 8k (1 - tan^2 4k)

then if I'm doing this correctly

tan 8k (1 - tan 2k) (1 + tan 2k)

But from here I'm stuck, that is if I'm on the right track.
 
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  • #2
"tan 8k (1 - tan^2 4k)"

From here use the double-angle formula.
 
  • #3
tan 8k (1 - tan 4k) (1 + tan 4k), this is ok,although you did not need to factorize (1-tan^24k)at all.
now notice that we can write down tan8k=tan2*4k=2tan4k/(1-tan^2 4k), after substitution look what can be canceled out.
I am assuming that you already know why
tan2*4k=2tan4k/(1-tan^2 4k), is so, right??

Edit: like neutrino said, i just elaborated it a little bit more
 
Last edited:
  • #4
sutupidmath said:
tan 8k (1 - tan 2k) (1 + tan 2k), this is ok

It is not.
 
  • #5
neutrino said:
It is not.

Yeah, sorry by mistake, because i just copied this from the op and did not acyally look at it.!
What i meant was (1-tan4k)(1+tan4k).

sorry!
 
  • #6
It might also be a good idea not to lose track of the right side of the equation!
 

1. What does it mean to verify an equation as an identity?

Verifying an equation as an identity means to prove that the equation is true for all possible values of the variables involved.

2. How do you verify an equation as an identity?

To verify an equation as an identity, you need to manipulate the equation using algebraic properties and operations to show that both sides of the equation are equivalent.

3. What are some common techniques used to verify an equation as an identity?

Some common techniques used to verify an equation as an identity include substitution, simplification, and transforming one side of the equation to match the other side.

4. Can you give an example of verifying an equation as an identity?

Sure, for the equation sin^2(x) + cos^2(x) = 1, we can use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to show that both sides of the equation are equivalent.

5. Why is it important to verify an equation as an identity?

Verifying an equation as an identity is important because it ensures that the equation is true for all values of the variables involved, and not just for specific values. It also allows us to confidently use the equation in further calculations or proofs.

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