Understanding the Equality
To prove that cos⁴A + sin²A = sin⁴A + cos²A, we can start by rearranging the equation into a more manageable form.
Rearranging the Equation
- Move all terms to one side:
cos⁴A + sin²A - sin⁴A - cos²A = 0
- This simplifies to:
cos⁴A - sin⁴A + sin²A - cos²A = 0
Factoring the Expression
- Recognize that cos⁴A - sin⁴A can be factored using the difference of squares:
(cos²A + sin²A)(cos²A - sin²A)
- Since cos²A + sin²A = 1 (Pythagorean identity), we have:
1(cos²A - sin²A) + (sin²A - cos²A) = 0
Simplifying the Terms
- Notice that cos²A - sin²A and sin²A - cos²A are opposites:
(cos²A - sin²A) + (sin²A - cos²A) = 0
- This confirms that the left-hand side equals zero, validating the equation.
Conclusion
Thus, we have shown that:
cos⁴A + sin²A = sin⁴A + cos²A is indeed true based on fundamental trigonometric identities and algebraic manipulation.