How to simplify 1/(cos^4x+sin^4x)

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In summary, to simplify 1/(cos^4x+sin^4x), one can use the fundamental identity sin^2x + cos^2x = 1 and manipulate the equation to isolate sin^4x and cos^4x. This can be done using product identities and in the end, the equation can be rewritten as 1/sin^4x + cos^4x = something, which can then be substituted into the original equation.
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Homework Statement


Simplify 1/(cos^4x+sin^4x)


Homework Equations


One can't apply cosine of a difference when the cosine is raised to a power, right?


The Attempt at a Solution


I tried using several trig identities, but it doesn't get any simpler.

Thank you for any help in advance!
 
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  • #2
try replacing sin2x with 1-cos2x or the other way around.
 
  • #3
Try starting with the fundamental identity [tex]sin^{2}x + cos^{2}x=1[/tex]. Square both sides and foil. This might get messy, but if you keep all your terms in order you will be in good shape. Next, look for your [tex]sin^{4}x[/tex] and [tex]cos^{4}x[/tex] values. You want to isolate them on one side and have the other terms on the other side. Now, you have to go to work with the other trig identities on the rest of the terms. I was able to do it product identities. In the end, you should have [tex]sin^{4}x+cos^{4}x=something[/tex]. The last step is to plug this into your original equation [tex]1/sin^{4}x+cos^{4}x[/tex].
 
  • #4
Thanks for the help! I figured it out.
 

Related to How to simplify 1/(cos^4x+sin^4x)

1. How do you simplify 1/(cos^4x+sin^4x)?

To simplify 1/(cos^4x+sin^4x), we can use the identity cos^2x + sin^2x = 1 and rearrange the expression to get 1/(cos^2x + sin^2x)^2. Then, we can substitute 1 for cos^2x + sin^2x, giving us the simplified form of 1/1^2, which is equal to 1.

2. Can you simplify 1/(cos^4x+sin^4x) further?

No, 1/(cos^4x+sin^4x) is already in its simplest form.

3. Is there a way to rewrite 1/(cos^4x+sin^4x) in terms of trigonometric functions?

Yes, we can rewrite 1/(cos^4x+sin^4x) as sec^4x/(1+tan^4x) using the identities cos^2x = sec^2x - 1 and sin^2x = tan^2x + 1. However, this form is not considered simpler than the original expression.

4. How does simplifying 1/(cos^4x+sin^4x) help in solving trigonometric equations?

Simplifying 1/(cos^4x+sin^4x) can help in solving trigonometric equations by reducing the complexity of the expression, making it easier to manipulate and solve. This can also help in identifying patterns and relationships between different trigonometric functions.

5. Can you simplify 1/(cos^4x+sin^4x) using a calculator?

No, most calculators do not have a simplification function for trigonometric expressions. However, they can evaluate the expression for specific values of x.

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