Trigonometric Identities- simplify sin^4x - cos^4x

In summary, the conversation discusses a method for simplifying and proving the trigonometric identity sin^4x-cos^4x = 1-2cos^2x. The method involves factoring and using the Pythagorean identity to simplify before combining like terms and multiplying to obtain the desired result. The conversation also mentions using theta instead of x and expresses appreciation for any ideas or assistance.
  • #1
ku1005
66
0
hey just wonderin if any1 could give me a hint as to the best method to prove the following trigonometric indentity:

sin^4x-cos^4x = 1 - 2cos^2x

i tried the side more complicated first...but can't seem to hav any luck...other then maing it more complicated!

umm the x's are meant to be thetas...but didnlt ahv the symbol handy.and its meant to be read sign to he power 4 theta...really aprreciate any ideas for this one!
 
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  • #2
sorry got guys!...kool just factor out as 2 lots of sin^2theta...use pythagorean indentity to sub and then cancel like terms...WORKS a charm lol...thanks if anyoe read tis anyways!
 
  • #3
got it guys*
 
  • #4
How To Simplify and Prove: sin^4x-cos^4x

Given: sin^4x-cos^4x

(sin^2x-cos^2x)(sin^2x+cos^2x) Factoring
(1)(sin^2x-cos^2x) Pythagorean Identity
(1)(1-cos^2x-cos^2x) Pythagorean Identity
(1)(1-2cos^2x) Combine Like Terms
1-2cos^2x Multiply


Hope This Helps. :approve:
 
  • #5
LadyJ123 said:
Given: sin^4x-cos^4x

(sin^2x-cos^2x)(sin^2x+cos^2x) Factoring
(1)(sin^2x-cos^2x) Pythagorean Identity
(1)(1-cos^2x-cos^2x) Pythagorean Identity
(1)(1-2cos^2x) Combine Like Terms
1-2cos^2x Multiply


Hope This Helps. :approve:

Nice job LadyJ123. But in the future, don't post complete solutions to problems. Just give hints. It's in the forum rules.
 

FAQ: Trigonometric Identities- simplify sin^4x - cos^4x

What is a trigonometric identity?

A trigonometric identity is a mathematical equation that involves trigonometric functions (such as sine, cosine, tangent) and is true for all values of the variables.

How do you simplify the expression sin^4x - cos^4x?

The expression can be simplified using the trigonometric identity: sin^2x = 1 - cos^2x. By substituting this identity into the original expression, we get: sin^4x - cos^4x = (sin^2x)^2 - cos^4x = (1 - cos^2x)^2 - cos^4x. This can then be simplified further using algebraic techniques.

Can trigonometric identities be proven?

Yes, trigonometric identities can be proven using algebraic techniques and geometric reasoning. They can also be verified using the unit circle and other methods.

How are trigonometric identities used in mathematics?

Trigonometric identities are used in various fields of mathematics, such as calculus, geometry, and physics. They are useful for simplifying equations, solving trigonometric equations, and making connections between different trigonometric functions.

Are there any common mistakes to avoid when using trigonometric identities?

Yes, some common mistakes to avoid include not using the correct identity for a given problem, making errors in algebraic simplification, and forgetting to check for extraneous solutions. It is important to carefully follow the steps and double check your work when using trigonometric identities.

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