Differentiate and simplify: f(x)=sin^2(2x)-cos^2(2x)

In summary, the conversation discusses whether or not a given expression has been fully simplified and if there exists a simpler expression. The conclusion is that the expression is fully simplified and the simplest path to the simplest result has been taken.
  • #1
ttpp1124
110
4
Homework Statement
Has this been simplified fully?
Relevant Equations
n/a
Screen Shot 2020-05-11 at 10.11.17 PM.png
 
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  • #3
yes
 
  • #4
Is that an answer to the quesion in post #1 or the one in post #2 ? Or both ? :cool:

Let me ask @ttpp1124 : do you expect there exists a simpler expression ? In what way ?
 
Last edited:
  • #5
yes, it is fully simplified, and of course with trig you could go in many circles with identities but I believe the one stopped at is simplest.
 
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  • #6
Mark44 said:
Do you have a question here?
yes, sorry, I forgot to include the question! here it is:
 

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  • #7
I would use [itex]\cos (2\theta) = \cos^2 \theta - \sin^2\theta[/itex], from which immediately [itex]y = -\cos(4x)[/itex].
 
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  • #8
ttpp1124 said:
yes, sorry, I forgot to include the question! here it is:
Not much of a question !

Kudos @pasmith : the simplest path to the simplest result !
 

1. What is the purpose of differentiating and simplifying a function?

Differentiating a function allows us to find the rate of change of that function at a specific point. Simplifying a function makes it easier to understand and work with, and can reveal important properties of the function.

2. How do you differentiate a function?

To differentiate a function, we use the rules of differentiation, such as the power rule, product rule, and chain rule. These rules allow us to find the derivative of a function at a specific point.

3. What is the derivative of sin^2(2x)?

The derivative of sin^2(2x) is 4sin(2x)cos(2x). This can be found using the chain rule, where we first take the derivative of the outer function, sin^2(2x), and then multiply it by the derivative of the inner function, 2x.

4. How do you simplify f(x)=sin^2(2x)-cos^2(2x)?

To simplify this function, we can use the trigonometric identity sin^2(x) + cos^2(x) = 1. This allows us to rewrite the function as f(x) = sin^2(2x) + sin^2(2x) - 1. Then, using the power rule, we can simplify further to f(x) = 2sin^2(2x) - 1.

5. What is the final simplified form of f(x)=sin^2(2x)-cos^2(2x)?

The final simplified form of f(x) is f(x) = 2sin^2(2x) - 1. This form is easier to work with and can reveal important properties of the function, such as the maximum and minimum values.

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