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

• ttpp1124
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.
ttpp1124
Homework Statement
Has this been simplified fully?
Relevant Equations
n/a

ttpp1124 said:
Homework Statement:: Has this been simplified fully?
Relevant Equations:: n/a

View attachment 262618
Do you have a question here?

yes

Is that an answer to the quesion in post #1 or the one in post #2 ? Or both ?

Let me ask @ttpp1124 : do you expect there exists a simpler expression ? In what way ?

Last edited:
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.

BvU
Mark44 said:
Do you have a question here?
yes, sorry, I forgot to include the question! here it is:

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I would use $\cos (2\theta) = \cos^2 \theta - \sin^2\theta$, from which immediately $y = -\cos(4x)$.

BvU and PeroK
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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